New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Neighbor As Hyper Nebbish On Super Nebulous

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\fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
\fancyfoot[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
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\textbf\newline{
New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Neighbor As Hyper Nebbish On Super Nebulous
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Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA
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\section{ABSTRACT}
In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a {\tiny Hamiltonian-Cycle-Neighbor} pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if the following expression is called Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria holds
  \begin{eqnarray*}
&&
\forall N(E_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}
 Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if the following expression is called Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria holds
  \begin{eqnarray*}
&&
\forall N(E_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}
 and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  if the following expression is called Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria holds
  \begin{eqnarray*}
&&
\forall N(V_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}
  Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if the following expression is called Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria holds
  \begin{eqnarray*}
&&
\forall N(V_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}
and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. ((Neutrosophic) SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}).
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called an Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; an Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial  if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; a Neutrosophic V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; an Extreme V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.  In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  and Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Two different types of SuperHyperDefinitions are debut for them but the research goes further and the SuperHyperNotion, SuperHyperUniform, and  SuperHyperClass based on that are well-defined and well-reviewed. The literature review is implemented in the whole of this research. For shining the elegancy and the significancy of this research, the comparison between this SuperHyperNotion with other SuperHyperNotions and fundamental SuperHyperNumbers are featured. The definitions are followed by the examples and the instances thus the clarifications are driven with different tools. The applications are figured out to make sense about the theoretical aspect of this ongoing research. The ``Cancer's Recognition'' are the under research to figure out the challenges make sense about ongoing and upcoming research. The special case is up. The cells are viewed in the deemed ways. There are different types of them. Some of them are individuals and some of them are well-modeled by the group of cells. These types are all officially called ``SuperHyperVertex'' but the relations amid them all officially called ``SuperHyperEdge''. The frameworks ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph'' are chosen and elected to research about ``Cancer's Recognition''. Thus these complex and dense SuperHyperModels open up some avenues to research on theoretical segments and ``Cancer's Recognition''. Some avenues are posed to pursue this research. It's also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph. Assume a SuperHyperGraph. Then $\delta-$SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is a maximal    of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (Neutrosophic) cardinalities of  SuperHyperNeighbors of $s\in S:$ there are $|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta;$ and $ |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$
The first Expression, holds if $S$ is an $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an $\delta-$SuperHyperDefensive; a Neutrosophic $\delta-$SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is a maximal Neutrosophic     of SuperHyperVertices with maximum Neutrosophic cardinality such that either of the following expressions hold for the Neutrosophic cardinalities of  SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{Neutrosophic} > |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta;$ and
$ |S\cap N(s)|_{Neutrosophic} < |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta.$
The first Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperDefensive
  It's useful to define a ``Neutrosophic'' version of a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . Since there's more ways to get type-results to make a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  more understandable. For the sake of having Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, there's a need to ``redefine'' the notion of a ``SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . It's redefined a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  if the mentioned Table holds, concerning, ``The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph'' with the key points,  
``The Values of The Vertices \& The Number of Position in Alphabet'',
``The Values of The SuperVertices\&The maximum Values of Its Vertices'',
``The Values of The Edges\&The maximum Values of Its Vertices'',
``The Values of The HyperEdges\&The maximum Values of Its Vertices'', ``The Values of The SuperHyperEdges\&The maximum Values of Its Endpoints''. To get structural examples and instances, I'm going to introduce the next SuperHyperClass of SuperHyperGraph based on a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . It's the main. It'll be disciplinary to have the foundation of previous definition in the kind of SuperHyperClass. If there's a need to have all SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} until the SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then it's officially called a ``SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' but otherwise, it isn't a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . There are some instances about the clarifications for the main definition titled a ``SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . For the sake of having a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' and a ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a Neutrosophic SuperHyperGraph. It's redefined ``Neutrosophic SuperHyperGraph'' if the intended Table holds. And a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  are redefined to a ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' if the intended Table holds. It's useful to define ``Neutrosophic'' version of SuperHyperClasses. Since there's more ways to get Neutrosophic type-results to make a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended Table holds. Thus  SuperHyperPath,  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
``Neutrosophic SuperHyperPath'', ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'', ``Neutrosophic SuperHyperStar'', ``Neutrosophic SuperHyperBipartite'', ``Neutrosophic SuperHyperMultiPartite'', and ``Neutrosophic SuperHyperWheel'' if the intended Table holds.  A SuperHyperGraph has a ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' where it's the strongest [the maximum Neutrosophic value from all the SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  amid the maximum value amid all SuperHyperVertices from a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} .] SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} . A graph is a SuperHyperUniform if it's a SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume a Neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows. It's SuperHyperPath if it's only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions; it's SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if it's only one SuperVertex as intersection amid two given SuperHyperEdges; it's SuperHyperStar it's only one SuperVertex as intersection amid all SuperHyperEdges;
it's SuperHyperBipartite it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming two separate sets, has no SuperHyperEdge in common; it's SuperHyperMultiPartite  it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no SuperHyperEdge in common; it's a SuperHyperWheel if it's only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex. The SuperHyperModel proposes the specific designs and the specific architectures. The SuperHyperModel is officially called ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph''. In this SuperHyperModel, The ``specific'' cells and ``specific group'' of cells are SuperHyperModeled as ``SuperHyperVertices'' and the common and intended properties between ``specific'' cells and ``specific group'' of cells are SuperHyperModeled as ``SuperHyperEdges''. Sometimes, it's useful to have some degrees of determinacy, indeterminacy, and neutrality to have more precise SuperHyperModel which in this case the SuperHyperModel is called ``Neutrosophic''. In the future research, the foundation will be based on the ``Cancer's Recognition'' and the results and the definitions will be introduced in redeemed ways. The recognition of the cancer in the long-term function. The specific region has been assigned by the model [it's called SuperHyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be Neutrosophic SuperHyperGraph] to have convenient perception on what's happened and what's done. There are some specific models, which are well-known and they've got the names, and some SuperHyperGeneral SuperHyperModels. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a Neutrosophic SuperHyperPath(-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  or the strongest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}  in those Neutrosophic SuperHyperModels. For the longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, called SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and the strongest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, called Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, some general results are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths have only two SuperHyperEdges but it's not enough since it's essential to have at least three SuperHyperEdges to form any style of a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. There isn't any formation of any SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} but literarily, it's the deformation of any SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. It, literarily, deforms and it doesn't form.  A basic familiarity with Neutrosophic  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.
\\ \vspace{4mm}
\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Cancer's Neutrosophic Recognition
   \\
\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45
\section{Applied Notions Under The Scrutiny Of The Motivation Of This Scientific Research}
In this scientific research, there are some ideas in the featured frameworks of motivations. I try to bring the motivations in the narrative ways. Some cells have been faced with some attacks from the situation which is caused by the cancer's attacks. In this case, there are some embedded analysis on the ongoing situations which in that, the cells could be labelled as some groups and some groups or individuals have excessive labels which all are raised from the behaviors to overcome the cancer's attacks. In the embedded situations, the individuals of cells and the groups of cells could be considered as ``new groups''. Thus it motivates us to find the proper SuperHyperModels for getting more proper analysis on this messy story. I've found the SuperHyperModels which are officially called ``SuperHyperGraphs'' and ``Neutrosophic SuperHyperGraphs''. In this SuperHyperModel, the cells and the groups of cells are defined as ``SuperHyperVertices'' and the relations between the individuals of cells and the groups of cells are defined as ``SuperHyperEdges''. Thus it's another motivation for us to do research on this SuperHyperModel based on the ``Cancer's Recognition''. Sometimes, the situations get worst. The situation is passed from the certainty and precise style. Thus it's the beyond them. There are three descriptions, namely, the degrees of determinacy, indeterminacy and neutrality, for any object based on vague forms, namely, incomplete data, imprecise data, and uncertain analysis. The latter model could be considered on the previous SuperHyperModel. It's SuperHyperModel. It's SuperHyperGraph but it's officially called ``Neutrosophic SuperHyperGraphs''. The cancer is the disease but the model is going to figure out what's going on this phenomenon. The special case of this disease is considered and as the consequences of the model, some parameters are used. The cells are under attack of this disease but the moves of the cancer in the special region are the matter of mind. The recognition of the cancer could help to find some treatments for this disease. The SuperHyperGraph and Neutrosophic SuperHyperGraph are the SuperHyperModels on the ``Cancer's Recognition'' and both bases are the background of this research. Sometimes the cancer has been happened on the region, full of cells, groups of cells and embedded styles. In this segment, the SuperHyperModel proposes some SuperHyperNotions based on the connectivities of the moves of the cancer in the forms of alliances' styles with the formation of the design and the architecture are formally called ``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' in the themes of jargons and buzzwords. The prefix ``SuperHyper'' refers to the theme of the embedded styles to figure out the background for the SuperHyperNotions.  The recognition of the cancer in the long-term function. The specific region has been assigned by the model [it's called SuperHyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be Neutrosophic SuperHyperGraph] to have convenient perception on what's happened and what's done. There are some specific models, which are well-known and they've got the names, and some general models. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a Neutrosophic SuperHyperPath (-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the optimal   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} or the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in those Neutrosophic SuperHyperModels. Some general results are introduced. Beyond that in SuperHyperStar, all possible Neutrosophic SuperHyperPath s have only two SuperHyperEdges but it's not enough since it's essential to have at least three SuperHyperEdges to form any style of a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. There isn't any formation of any SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} but literarily, it's the deformation of any SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. It, literarily, deforms and it doesn't form.
\begin{question}
How to define the SuperHyperNotions and to do research on them to find the `` amount of   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' of either individual of cells or the groups of cells based on the fixed cell or the fixed group of cells, extensively, the ``amount of   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' based on the fixed groups of cells or the fixed groups of group of cells?
\end{question}
\begin{question}
What are the best descriptions for the ``Cancer's Recognition'' in terms of these messy and dense SuperHyperModels where embedded notions are illustrated?
\end{question}
It's motivation to find notions to use in this dense model is titled ``SuperHyperGraphs''. Thus it motivates us to define different types of ``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' and ``Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' on ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph''. Then the research has taken more motivations to define SuperHyperClasses and to find some connections amid this SuperHyperNotion with other SuperHyperNotions. It motivates us to get some instances and examples to make clarifications about the framework of this research. The general results and some results about some connections are some avenues to make key point of this research, ``Cancer's Recognition'', more understandable and more clear.
\\
The framework of this research is as follows. In the beginning, I introduce basic definitions to clarify about preliminaries. In the subsection ``Preliminaries'', initial definitions about SuperHyperGraphs and Neutrosophic SuperHyperGraph are deeply-introduced and in-depth-discussed. The elementary concepts are clarified and illustrated completely and sometimes review literature are applied to make sense about what's going to figure out about the upcoming sections. The main definitions and their clarifications alongside some results about new notions,   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, are figured out in sections ``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' and ``Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}''. In the sense of tackling on getting results and in {\tiny Hamiltonian-Cycle-Neighbor} to make sense about continuing the research, the ideas of SuperHyperUniform and Neutrosophic SuperHyperUniform are introduced and as their consequences, corresponded SuperHyperClasses are figured out to debut what's done in this section, titled ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. As going back to origin of the notions, there are some smart steps toward the common notions to extend the new notions in new frameworks, SuperHyperGraph and Neutrosophic SuperHyperGraph, in the sections ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. The starter research about the general SuperHyperRelations and as concluding and closing section of theoretical research are contained in the section ``General Results''. Some general SuperHyperRelations are fundamental and they are well-known as fundamental SuperHyperNotions as elicited and discussed in the sections, ``General Results'',
``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'', ``Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'', ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. There are curious questions about what's done about the SuperHyperNotions to make sense about excellency of this research and going to figure out the word ``best'' as the description and adjective for this research as presented in section, ``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}''. The keyword of this research debut in the section ``Applications in Cancer's Recognition'' with two cases and subsections ``Case 1: The Initial Steps Toward SuperHyperBipartite as SuperHyperModel'' and ``Case 2: The Increasing Steps Toward SuperHyperMultipartite as SuperHyperModel''. In the section, ``Open Problems'', there are some scrutiny and discernment on what's done and what's happened in this research in the terms of ``questions'' and ``problems'' to make sense to figure out this research in featured style. The advantages and the limitations of this research alongside about what's done in this research to make sense  and to get sense about what's figured out are included in the section, ``Conclusion and Closing Remarks''.
\section{Neutrosophic Preliminaries Of This Scientific Research On the Redeemed Ways}
In this section, the basic material in this scientific research, is referred to [Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2), [Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1), [Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.5,p.2), [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [t-norm](\textbf{Ref.}\cite{HG38}, Definition 2.7, p.3), and [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [Neutrosophic Strength of the Neutrosophic SuperHyperPaths] (\textbf{Ref.}\cite{HG38},Definition 5.3,p.7), and [Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)] (\textbf{Ref.}\cite{HG38},Definition 5.4,p.7). Also, the new ideas and their clarifications are addressed to \textbf{Ref.}\cite{HG38}.
\\
In this subsection, the basic material which is used in this scientific research, is presented. Also, the new ideas and their clarifications are elicited.
\begin{definition}[Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1).\\
 Let $X$ be a {\tiny Hamiltonian-Cycle-Neighbor} of points (objects) with generic elements in $X$ denoted by $x;$ then the \textbf{Neutrosophic set} $A$ (NS $A$) is an object having the form
$$A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$$
where the functions $T, I, F: X\rightarrow]^-0,1\leftidx{^+}{[}$ define respectively the a \textbf{truth-membership function}, an \textbf{indeterminacy-membership function}, and a \textbf{falsity-membership function} of the element $x\in X$ to the set $A$ with the condition
$$\leftidx{^-}{0} \leq T_A(x)+I_A(x)+F_A(x)\leq 3^{+}.$$
The functions $T_A(x),I_A(x)$ and $F_A(x)$ are real standard or nonstandard subsets of $]^-0,1\leftidx{^+}{[}.$
\end{definition}
\begin{definition}[Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2).\\
 Let $X$ be a {\tiny Hamiltonian-Cycle-Neighbor} of points (objects) with generic elements in $X$ denoted by $x.$ A \textbf{single valued Neutrosophic set} $A$ (SVNS $A$) is characterized by truth-membership function $T_A(x),$ an indeterminacy-membership function $I_A(x),$ and a falsity-membership function $F_A(x).$ For each point $x$ in $X,$ $T_A(x),I_A(x),F_A(x)\in [0,1].$ A SVNS $A$ can be written as
 $$A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}.$$
\end{definition}
\begin{definition}
The \textbf{degree of truth-membership}, \textbf{indeterminacy-membership} and \textbf{falsity-membership of the subset} $X\subset A$ of the single valued Neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:$$T_A(X)=\min[T_A(v_i),T_A(v_j)]_{v_i,v_j\in X},$$
$$I_A(X)=\min[I_A(v_i),I_A(v_j)]_{v_i,v_j\in X},$$
$$ \text{and}~F_A(X)=\min[F_A(v_i),F_A(v_j)]_{v_i,v_j\in X}.$$
\end{definition}
\begin{definition}
The \textbf{support} of $X\subset A$ of the single valued Neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:
$$supp(X)=\{x\in X:~T_A(x),I_A(x),F_A(x)> 0\}.$$
\end{definition}
\begin{definition}[Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.5,p.2).\\
Assume $V'$ is a given set. a \textbf{Neutrosophic SuperHyperGraph} (NSHG) $S$
is a pair $S=(V,E),$ where
\begin{itemize}
\item[$(i)$] $V=\{V_1,V_2,\ldots,V_n\}$ a finite set of finite single valued Neutrosophic subsets of $V';$
\item[$(ii)$] $V=\{(V_i,T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)):~T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)\geq0\},~(i=1,2,\ldots,n);$
\item[$(iii)$] $E=\{E_1,E_2,\ldots,E_{n'}\}$ a finite set of finite single valued Neutrosophic subsets of $V;$
\item[$(iv)$] $E=\{(E_{i'},T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})):~T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})\geq0\},~(i'=1,2,\ldots,n');$
\item[$(v)$] $V_i\neq\emptyset,~(i=1,2,\ldots,n);$
\item[$(vi)$] $E_{i'}\neq\emptyset,~(i'=1,2,\ldots,n');$
\item[$(vii)$] $\sum_{i}supp(V_i)=V,~(i=1,2,\ldots,n);$
\item[$(viii)$] $\sum_{i'}supp(E_{i'})=V,~(i'=1,2,\ldots,n');$   
\item[$(ix)$] and the following conditions hold:
$$T'_V(E_{i'})\leq\min[T_{V'}(V_i),T_{V'}(V_j)]_{V_i,V_j\in E_{i'}},$$
$$ I'_V(E_{i'})\leq\min[I_{V'}(V_i),I_{V'}(V_j)]_{V_i,V_j\in E_{i'}},$$
$$ \text{and}~F'_V(E_{i'})\leq\min[F_{V'}(V_i),F_{V'}(V_j)]_{V_i,V_j\in E_{i'}}$$
where $i'=1,2,\ldots,n'.$
\end{itemize}
Here the Neutrosophic SuperHyperEdges (NSHE) $E_{j'}$ and the Neutrosophic SuperHyperVertices (NSHV) $V_j$ are single valued Neutrosophic sets. $T_{V'}(V_i),I_{V'}(V_i),$ and $F_{V'}(V_i)$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership the Neutrosophic SuperHyperVertex (NSHV) $V_i$ to the Neutrosophic SuperHyperVertex (NSHV) $V.$ $T'_{V}(E_{i'}),T'_{V}(E_{i'}),$ and $T'_{V}(E_{i'})$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership of the Neutrosophic SuperHyperEdge (NSHE) $E_{i'}$ to the Neutrosophic SuperHyperEdge (NSHE) $E.$ Thus, the $ii'$th element of the \textbf{incidence matrix} of Neutrosophic SuperHyperGraph (NSHG) are of the form $(V_i,T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'}))$, the sets V and E are crisp sets.
\end{definition}
\begin{definition}[Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ The Neutrosophic SuperHyperEdges (NSHE) $E_{i'}$ and the Neutrosophic SuperHyperVertices (NSHV) $V_i$ of Neutrosophic SuperHyperGraph (NSHG) $S=(V,E)$ could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If $|V_i|=1,$ then $V_i$ is called \textbf{vertex};
\item[$(ii)$] if $|V_i|\geq1,$ then $V_i$ is called \textbf{SuperVertex};
\item[$(iii)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{edge};
\item[$(iv)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{HyperEdge};
\item[$(v)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{SuperEdge};
\item[$(vi)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{SuperHyperEdge}.
\end{itemize}
\end{definition}
If we choose different types of binary operations, then we could get hugely diverse types of general forms of Neutrosophic SuperHyperGraph (NSHG).
\begin{definition}[t-norm](\textbf{Ref.}\cite{HG38}, Definition 2.7, p.3).\\
 A binary operation $\otimes: [0, 1] \times [0, 1] \rightarrow [0, 1]$ is a \textbf{$t$-norm} if it satisfies the following for $x,y,z,w  \in [0, 1]$:
\begin{itemize}
\item[$(i)$] $1 \otimes x =x;$
\item[$(ii)$] $x \otimes y = y \otimes x;$
\item[$(iii)$] $x \otimes (y \otimes z) = (x \otimes y) \otimes z;$
\item[$(iv)$] If $ w \leq x$ and $y \leq z$ then $w \otimes y \leq x \otimes z.$
\end{itemize}
\end{definition}
\begin{definition}
The \textbf{degree of truth-membership}, \textbf{indeterminacy-membership} and \textbf{falsity-membership of the subset} $X\subset A$ of the single valued Neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$ (with respect to t-norm $T_{norm}$):$$T_A(X)=T_{norm}[T_A(v_i),T_A(v_j)]_{v_i,v_j\in X},$$
$$I_A(X)=T_{norm}[I_A(v_i),I_A(v_j)]_{v_i,v_j\in X},$$
$$ \text{and}~F_A(X)=T_{norm}[F_A(v_i),F_A(v_j)]_{v_i,v_j\in X}.$$
\end{definition}
\begin{definition}
The \textbf{support} of $X\subset A$ of the single valued Neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:
$$supp(X)=\{x\in X:~T_A(x),I_A(x),F_A(x)> 0\}.$$
\end{definition}
\begin{definition}(General Forms of  Neutrosophic SuperHyperGraph (NSHG)).\\
Assume $V'$ is a given set. a \textbf{Neutrosophic SuperHyperGraph} (NSHG) $S$
is a pair $S=(V,E),$ where
\begin{itemize}
\item[$(i)$] $V=\{V_1,V_2,\ldots,V_n\}$ a finite set of finite single valued Neutrosophic subsets of $V';$
\item[$(ii)$] $V=\{(V_i,T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)):~T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)\geq0\},~(i=1,2,\ldots,n);$
\item[$(iii)$] $E=\{E_1,E_2,\ldots,E_{n'}\}$ a finite set of finite single valued Neutrosophic subsets of $V;$
\item[$(iv)$] $E=\{(E_{i'},T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})):~T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})\geq0\},~(i'=1,2,\ldots,n');$
\item[$(v)$] $V_i\neq\emptyset,~(i=1,2,\ldots,n);$
\item[$(vi)$] $E_{i'}\neq\emptyset,~(i'=1,2,\ldots,n');$
\item[$(vii)$] $\sum_{i}supp(V_i)=V,~(i=1,2,\ldots,n);$
\item[$(viii)$] $\sum_{i'}supp(E_{i'})=V,~(i'=1,2,\ldots,n').$   
\end{itemize}
Here the Neutrosophic SuperHyperEdges (NSHE) $E_{j'}$ and the Neutrosophic SuperHyperVertices (NSHV) $V_j$ are single valued Neutrosophic sets. $T_{V'}(V_i),I_{V'}(V_i),$ and $F_{V'}(V_i)$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership the Neutrosophic SuperHyperVertex  (NSHV) $V_i$ to the Neutrosophic SuperHyperVertex (NSHV) $V.$ $T'_{V}(E_{i'}),T'_{V}(E_{i'}),$ and $T'_{V}(E_{i'})$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership of the Neutrosophic SuperHyperEdge (NSHE) $E_{i'}$ to the Neutrosophic SuperHyperEdge (NSHE) $E.$ Thus, the $ii'$th element of the \textbf{incidence matrix} of Neutrosophic SuperHyperGraph (NSHG) are of the form $(V_i,T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'}))$, the sets V and E are crisp sets.
\end{definition}

\begin{definition}[Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ The Neutrosophic SuperHyperEdges (NSHE) $E_{i'}$ and the Neutrosophic SuperHyperVertices (NSHV) $V_i$ of Neutrosophic SuperHyperGraph (NSHG) $S=(V,E)$ could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If $|V_i|=1,$ then $V_i$ is called \textbf{vertex};
\item[$(ii)$] if $|V_i|\geq1,$ then $V_i$ is called \textbf{SuperVertex};
\item[$(iii)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{edge};
\item[$(iv)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{HyperEdge};
\item[$(v)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{SuperEdge};
\item[$(vi)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{SuperHyperEdge}.
\end{itemize}
\end{definition}
This SuperHyperModel is too messy and too dense. Thus there's a need to have some restrictions and conditions on SuperHyperGraph. The special case of this SuperHyperGraph makes the patterns and regularities.
\begin{definition}
 A graph is \textbf{SuperHyperUniform} if it's SuperHyperGraph and the number of elements of SuperHyperEdges are the same.
\end{definition}
To get more visions on SuperHyperUniform, the some SuperHyperClasses are introduced. It makes to have SuperHyperUniform more understandable.
\begin{definition}
 Assume a Neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows.
\begin{itemize}
 \item[(i).] It's \textbf{Neutrosophic SuperHyperPath } if it's only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions;
 \item[(ii).] it's \textbf{SuperHyperCycle} if it's only one SuperVertex as intersection amid two given SuperHyperEdges;
  \item[(iii).] it's \textbf{SuperHyperStar} it's only one SuperVertex as intersection amid all SuperHyperEdges;
   \item[(iv).] it's \textbf{SuperHyperBipartite} it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming two separate sets, has no SuperHyperEdge in common;
    \item[(v).] it's \textbf{SuperHyperMultiPartite}  it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no SuperHyperEdge in common;
     \item[(vi).] it's \textbf{SuperHyperWheel} if it's only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex.

\end{itemize}
\end{definition}

\begin{definition}
 Let a pair $S=(V,E)$ be a Neutrosophic SuperHyperGraph (NSHG) $S.$ Then a sequence of Neutrosophic SuperHyperVertices (NSHV) and Neutrosophic SuperHyperEdges (NSHE)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s$$
  is called a \textbf{Neutrosophic  SuperHyperPath } (NSHP) from Neutrosophic SuperHyperVertex  (NSHV) $V_1$ to Neutrosophic SuperHyperVertex (NSHV) $V_s$ if either of following conditions hold:
  \begin{itemize}
 \item[$(i)$]  $V_i,V_{i+1}\in E_{i'};$
 \item[$(ii)$]   there's a vertex $v_i\in V_i$ such that $v_i,V_{i+1}\in E_{i'};$
  \item[$(iii)$]  there's a SuperVertex $V'_i \in V_i$ such that $V'_i,V_{i+1}\in E_{i'};$
   \item[$(iv)$]   there's a vertex $v_{i+1}\in V_{i+1}$ such that $V_i,v_{i+1}\in E_{i'};$
  \item[$(v)$]  there's a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $V_i,V'_{i+1}\in E_{i'};$
   \item[$(vi)$]   there are a vertex $v_i\in V_i$ and a vertex $v_{i+1}\in V_{i+1}$ such that $v_i,v_{i+1}\in E_{i'};$
  \item[$(vii)$]   there are a vertex $v_i\in V_i$ and a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $v_i,V'_{i+1}\in E_{i'};$  
     \item[$(viii)$]   there are a SuperVertex $V'_i\in V_i$ and a vertex $v_{i+1}\in V_{i+1}$ such that $V'_i,v_{i+1}\in E_{i'};$
  \item[$(ix)$]   there are a SuperVertex $V'_i\in V_i$ and a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $V'_i,V'_{i+1}\in E_{i'}.$  
\end{itemize}
\end{definition}

\begin{definition}(Characterization of the Neutrosophic  SuperHyperPaths).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ a Neutrosophic  SuperHyperPath  (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_1$ to Neutrosophic SuperHyperVertex (NSHV) $V_s$ is sequence of Neutrosophic SuperHyperVertices (NSHV) and Neutrosophic SuperHyperEdges (NSHE)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s,$$
could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If for all $V_i,E_{j'},$ $|V_i|=1,~|E_{j'}|=2,$ then NSHP is called \textbf{path};
\item[$(ii)$] if for all $E_{j'},$ $|E_{j'}|=2,$  and there's $V_i,$ $|V_i|\geq1,$ then NSHP is called \textbf{SuperPath};
\item[$(iii)$] if for all $V_i,E_{j'},$ $|V_i|=1,~|E_{j'}|\geq2,$ then NSHP is called \textbf{HyperPath};
\item[$(iv)$] if there are $V_i,E_{j'},$ $|V_i|\geq1,|E_{j'}|\geq2,$ then NSHP is called \textbf{Neutrosophic SuperHyperPath }.
\end{itemize}
\end{definition}
\begin{definition}[Neutrosophic Strength of the Neutrosophic SuperHyperPaths](\textbf{Ref.}\cite{HG38},Definition 5.3,p.7).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ A Neutrosophic SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_1$ to Neutrosophic SuperHyperVertex (NSHV) $V_s$ is sequence of Neutrosophic SuperHyperVertices (NSHV) and Neutrosophic SuperHyperEdges (NSHE)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s,$$
have
\begin{itemize}
 \item[$(i)$]\textbf{Neutrosophic t-strength} $(\min\{T(V_i)\},m,n)_{i=1}^s$;
\item[$(ii)$] \textbf{Neutrosophic i-strength} $(m,\min\{I(V_i)\},n)_{i=1}^s;$
\item[$(iii)$] \textbf{Neutrosophic f-strength} $(m,n,\min\{F(V_i)\})_{i=1}^s;$
\item[$(iv)$] \textbf{Neutrosophic strength} $(\min\{T(V_i)\},\min\{I(V_i)\},\min\{F(V_i)\})_{i=1}^s.$
\end{itemize}
\end{definition}

\begin{definition}[Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)](\textbf{Ref.}\cite{HG38},Definition 5.4,p.7).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a  pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
  \item[$(ix)$] \textbf{Neutrosophic t-connective} if $T(E)\geq$ maximum number of Neutrosophic t-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(x)$] \textbf{Neutrosophic i-connective} if $I(E)\geq$ maximum number of Neutrosophic i-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(xi)$] \textbf{Neutrosophic f-connective} if $F(E)\geq$ maximum number of Neutrosophic f-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(xii)$] \textbf{Neutrosophic connective} if $(T(E),I(E),F(E))\geq$ maximum number of Neutrosophic strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s.$
\end{itemize}
\end{definition}

 

\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if the following expression is called \textbf{Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria} holds
  \begin{eqnarray*}
&&
\forall N(E_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}

\item[$(ii)$] \textbf{Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if the following expression is called \textbf{Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria} holds
  \begin{eqnarray*}
&&
\forall N(E_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}

and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if the following expression is called \textbf{Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria} holds
  \begin{eqnarray*}
&&
\forall N(V_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
\end{eqnarray*}

\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if the following expression is called \textbf{Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} criteria} holds
  \begin{eqnarray*}
&&
\forall N(V_a)\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
 \text{the maximum number of  SuperHyperVertices};
 \end{eqnarray*}
and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
 \item[$(iii)$]
an \textbf{Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
 \item[$(vi)$]
a \textbf{Neutrosophic V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
 \item[$(vii)$]
an \textbf{Extreme V-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic re-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, Neutrosophic v-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and Neutrosophic rv-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{definition}((Extreme/Neutrosophic)$\delta-$SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$  Then
\begin{itemize}
 \item[$(i)$] an \textbf{$\delta-$SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} is a Neutrosophic kind of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} such that either of the following expressions hold for the Neutrosophic cardinalities of  SuperHyperNeighbors of $s\in S:$
\begin{eqnarray*}
&&|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta; \label{136EQN1}
 \\&& |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta. \label{136EQN2}
\end{eqnarray*}
The Expression \eqref{136EQN1}, holds if $S$ is an \textbf{$\delta-$SuperHyperOffensive}. And the Expression \eqref{136EQN2}, holds if $S$ is an \textbf{$\delta-$SuperHyperDefensive};
 \item[$(ii)$] a \textbf{Neutrosophic $\delta-$SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} is a Neutrosophic kind of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} such that either of the following Neutrosophic expressions hold for the Neutrosophic cardinalities of SuperHyperNeighbors of $s\in S:$
\begin{eqnarray*}
&&|S\cap N(s)|_{Neutrosophic} > |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta; \label{136EQN3}
 \\&& |S\cap N(s)|_{Neutrosophic} < |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta. \label{136EQN4}
\end{eqnarray*}
The Expression \eqref{136EQN3}, holds if $S$ is a \textbf{Neutrosophic $\delta-$SuperHyperOffensive}. And the Expression \eqref{136EQN4}, holds if $S$ is a \textbf{Neutrosophic $\delta-$SuperHyperDefensive}.
\end{itemize}
\end{definition}
For the sake of having a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, there's a need to ``\textbf{redefine}'' the notion of ``Neutrosophic SuperHyperGraph''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values.
\begin{definition}\label{136DEF1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$  It's redefined \textbf{Neutrosophic SuperHyperGraph} if the Table \eqref{136TBL3} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{136DEF1}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{136TBL3}
\end{table}
\end{definition}
It's useful to define a ``Neutrosophic'' version of SuperHyperClasses. Since there's more ways to get Neutrosophic type-results to make a Neutrosophic more understandable.
\begin{definition}\label{136DEF2}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ There are some \textbf{Neutrosophic SuperHyperClasses} if the Table \eqref{136TBL4} holds. Thus  Neutrosophic SuperHyperPath ,  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
 \textbf{Neutrosophic SuperHyperPath},  \textbf{Neutrosophic SuperHyperCycle}, \textbf{Neutrosophic SuperHyperStar}, \textbf{Neutrosophic SuperHyperBipartite}, \textbf{Neutrosophic SuperHyperMultiPartite}, and \textbf{Neutrosophic SuperHyperWheel} if the Table \eqref{136TBL4} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph, Mentioned in the Definition \eqref{136DEF2}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{136TBL4}
\end{table}
\end{definition}
It's useful to define a ``Neutrosophic'' version of a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Since there's more ways to get type-results to make a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} more Neutrosophicly understandable.
\\
For the sake of having a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, there's a need to ``\textbf{redefine}'' the Neutrosophic notion of ``Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values.
\begin{definition}\label{136DEF1}
 Assume a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. It's redefined a \textbf{Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}} if the Table \eqref{136TBL1} holds.  \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{136DEF1}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{136TBL1}
\end{table}
\end{definition}
\section{
Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Neutrosophic event).\\
  Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Any Neutrosophic k-subset of $A$ of $V$ is called \textbf{Neutrosophic k-event} and if $k=2,$ then Neutrosophic subset of $A$ of $V$ is called \textbf{Neutrosophic event}. The following expression is called \textbf{Neutrosophic probability} of $A.$
\begin{eqnarray}
 E(A)=\sum_{a\in A}E(a).
\end{eqnarray}
\end{definition}
\begin{definition}(Neutrosophic Independent).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. $s$ Neutrosophic k-events $A_i,~i\in I$ is called \textbf{Neutrosophic s-independent} if the following expression is called \textbf{Neutrosophic s-independent criteria}
\begin{eqnarray*}
 E(\cap_{i\in I}A_i)=\prod_{i\in I}P(A_i).
\end{eqnarray*}
And if $s=2,$ then Neutrosophic k-events of $A$ and $B$ is called \textbf{Neutrosophic independent}. The following expression is called \textbf{Neutrosophic independent criteria}
\begin{eqnarray}
 E(A\cap B)=P(A)P(B).
\end{eqnarray}
\end{definition}
\begin{definition}(Neutrosophic Variable).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Any k-function {\tiny Hamiltonian-Cycle-Neighbor} like $E$ is called \textbf{Neutrosophic k-Variable}. If $k=2$, then any 2-function {\tiny Hamiltonian-Cycle-Neighbor} like $E$ is called \textbf{Neutrosophic Variable}.
\end{definition}
The notion of independent on Neutrosophic Variable is likewise.
\begin{definition}(Neutrosophic Expectation).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. A Neutrosophic k-Variable $E$ has a number is called \textbf{Neutrosophic Expectation} if the following expression is called \textbf{Neutrosophic Expectation criteria}
\begin{eqnarray*}
 Ex(E)=\sum_{\alpha \in V}E(\alpha)P(\alpha).
\end{eqnarray*}
\end{definition}
\begin{definition}(Neutrosophic Crossing).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. A Neutrosophic number is called \textbf{Neutrosophic Crossing} if the following expression is called \textbf{Neutrosophic Crossing criteria}
\begin{eqnarray*}
 Cr(S)=\min\{\text{Number of Crossing in a Plane Embedding of } S\}.
\end{eqnarray*}
\end{definition}
\begin{lemma}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $m$ and $n$ propose special {\tiny Hamiltonian-Cycle-Neighbor}. Then with $m\geq 4n,$

\end{lemma}
\begin{proof}
 Consider a planar embedding $G$ of $G$ with $cr(G)$ crossings. Let $S$ be a Neutrosophic  random k-subset of $V$ obtained by choosing each SuperHyperVertex of $G$ Neutrosophic independently with probability {\tiny Hamiltonian-Cycle-Neighbor} $p := 4n/m,$ and set $H:=G[S]$ and $H:=G[S].$
 \\
Define random variables $X, Y, Z$ on $V$ as follows: $X$ is the Neutrosophic number of SuperHyperVertices, $Y$ the Neutrosophic number of SuperHyperEdges, and $Z$ the Neutrosophic number of crossings of $H.$ The trivial bound noted above, when applied to $H,$ yields the inequality $Z \geq cr(H) \geq Y-3X.$ By linearity of Neutrosophic Expectation,
$$E(Z) \geq E(Y )-3E(X).$$
Now $E(X) = pn,~E(Y ) = p^2m$ (each SuperHyperEdge having some SuperHyperEnds) and $E(Z) = p^4cr(G)$ (each crossing being defined by some SuperHyperVertices). Hence
$$p^4cr(G) \geq p^2m-3pn.$$
Dividing both sides by $p^4,$ we have:
 \begin{eqnarray*}
cr(G)\geq \frac{pm-3n}{p^3}=\frac{n}{{(4n/m)}^3}=\frac{1}{64}{m^3}{n^2}.
\end{eqnarray*}
\end{proof}
\begin{theorem}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $l$ be the Neutrosophic number of SuperHyperLines in the plane passing through at least $k+1$ of these points, where $1 \leq k \leq 2 \sqrt{2n}.$ Then $l<32n^2/k^3.$
\end{theorem}
\begin{proof}
Form a Neutrosophic SuperHyperGraph $G$ with SuperHyperVertex SuperHyperSet $P$ whose SuperHyperEdge are the segments between
conseNeighborive points on the SuperHyperLines which pass through at least $k+1$ points of $P.$ This
Neutrosophic SuperHyperGraph has at least $kl$ SuperHyperEdges and Neutrosophic crossing at most $l$ choose two. Thus either $kl < 4n,$ in which case $l < 4n/k \leq32n^2/k^3,$ or $l^2/2 > \text{l choose 2} \geq cr(G) \geq {(kl)}^3/64n^2$ by the Neutrosophic Crossing Lemma, and again $l < 32n^2/k^3.$
\end{proof}

\begin{theorem}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $k$ be the number of pairs of points of $P$ at unit SuperHyperDistance. Then $k < 5n^{4/3}.$
\end{theorem}
\begin{proof}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Draw a SuperHyperUnit SuperHyperCircle around each SuperHyperPoint of $P.$ Let $n_i$ be the Neutrosophic number of these SuperHyperCircles passing through exactly $i$ points of $P.$ Then  $\sum{i=0}^{n-1}n_i = n$ and $k = \frac{1}{2}\sum{i=0}^{n-1}in_i.$ Now form a Neutrosophic SuperHyperGraph $H$ with SuperHyperVertex SuperHyperSet $P$ whose SuperHyperEdges are the SuperHyperArcs between conseNeighborive SuperHyperPoints on the SuperHyperCircles that pass through at least three SuperHyperPoints
of $P.$ Then
 \begin{eqnarray*}
e(H)=\sum_{i=3}^{n-1}in_i=2k-n_1-2n_2\geq2k-2n.
\end{eqnarray*}
Some SuperHyperPairs of SuperHyperVertices of $H$ might be joined by some parallel SuperHyperEdges. Delete from $H$ one of each SuperHyperPair of parallel SuperHyperEdges, so as to obtain a simple Neutrosophic SuperHyperGraph $G$ with $e(G) \geq k-n.$ Now $cr(G)\leq n(n-1)$ because $G$ is formed from at most n SuperHyperCircles, and any two SuperHyperCircles cross at most twice. Thus either $e(G) < 4n,$ in which case $k < 5n < 5n^{4/3},$ or $n^2 > n(n-1) \geq cr(G) \geq {(k-n)}^3/64n^2$ by the Neutrosophic Crossing Lemma, and $k<4n^{4/3} +n<5n^{4/3}.$
\end{proof}
\begin{proposition}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $X$ be a nonnegative Neutrosophic Variable and t a positive real number. Then
  \begin{eqnarray*}
P(X\geq t) \leq \frac{E(X)}{t}.
\end{eqnarray*}
\end{proposition}
 \begin{proof}
   \begin{eqnarray*}
&& E(X)=\sum \{X(a)P(a):a\in V\} \geq \sum\{X(a)P(a):a\in V,X(a)\geq t\}
\\&&
\sum\{tP(a):a\in V,X(a)\geq t\}=t\sum\{P(a):a\in V,X(a)\geq t\}
\\&&
tP(X\geq t).
\end{eqnarray*}
Dividing the first and last members by $t$ yields the asserted inequality.
\end{proof}
 \begin{corollary}
  Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $X_n$ be a nonnegative integer-valued variable in a prob- ability {\tiny Hamiltonian-Cycle-Neighbor} $(V_n,E_n), n\geq1.$ If $E(X_n)\rightarrow0$ as $n\rightarrow \infty,$ then $P(X_n =0)\rightarrow1$ as $n \rightarrow \infty.$
\end{corollary}
 \begin{proof}
\end{proof}
 \begin{theorem}
  Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}.
 A special SuperHyperGraph in $G_{n,p}$ almost surely has stability number at most $\lceil2p^{-1} \log n\rceil.$
\end{theorem}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}.
 A special SuperHyperGraph in $G_{n,p}$ is up. Let $G\in \mathcal{G}_{n,p}$  and let $S$ be a given SuperHyperSet of $k+1$ SuperHyperVertices of $G,$ where $k\in \Bbb N.$ The probability that $S$ is a stable SuperHyperSet of $G$ is $(1-p)^{(k+1) \text{choose} 2},$ this being the probability that none of the $(k+1) \text{choose} 2$ pairs of SuperHyperVertices of $S$ is a SuperHyperEdge of the Neutrosophic SuperHyperGraph $G.$
 \\
Let $A_S$ denote the event that $S$ is a stable SuperHyperSet of $G,$ and let $X_S$ denote the indicator Neutrosophic Variable for this Neutrosophic Event. By equation, we have
  \begin{eqnarray*}
E(X_S) = P(X_S = 1) = P(A_S ) = (1-p)^{(k+1) \text{choose} 2}.
\end{eqnarray*}
Let $X$ be the number of stable SuperHyperSets of cardinality $k + 1$ in $G.$ Then
  \begin{eqnarray*}
X = \sum\{X_S : S \subseteq V, |S| = k + 1\}
\end{eqnarray*}
and so, by those,
\begin{eqnarray*}
E(X)= \sum\{E(X_S): S\subseteq V, |S|=k+1\}= (\text{n choose k+1})  (1-p)^{(k+1) \text{choose} 2}.
\end{eqnarray*}
We bound the right-hand side by invoking two elementary inequalities:
\begin{eqnarray*}
(\text{n choose k+1})\leq \frac{n^{k+1}}{(k+1)!} \text{and} 1-p\leq e^{-p}.
\end{eqnarray*}
This yields the following upper bound on $E(X).$
\begin{eqnarray*}
E(X)\leq\frac{n^{k+1}e^{-p)(k+1) \text{choose} 2}}{(k+1)!}=\frac{􏰇ne^{-pk/2}^{􏰈k+1}}{(k+1)!}
\end{eqnarray*}
Suppose now that $k = \lceil2p^{-1} \log n\rceil.$ Then $k \geq 2p^{-1} \log n,$ so $ne^{-pk/2} \leq 1.$   Because $k$ grows at least as fast as the logarithm of $n,$  implies that $E(X) \rightarrow 0$ as $n \rightarrow \infty.$ Because $X$ is integer-valued and nonnegative, we deduce from Corollary that $P (X = 0) \rightarrow 1$ as $n \rightarrow \infty.$ Consequently, a Neutrosophic SuperHyperGraph in $\mathcal{G}_{n,p}$ almost surely has stability number at most $k.$
\end{proof}
\begin{definition}(Neutrosophic Variance).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. A Neutrosophic k-Variable $E$ has a number is called \textbf{Neutrosophic Variance} if the following expression is called \textbf{Neutrosophic Variance criteria}
\begin{eqnarray*}
 Vx(E)=Ex({(X-Ex(X))}^2).
\end{eqnarray*}
\end{definition}
 \begin{theorem}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $X$ be a Neutrosophic Variable and let $t$ be a positive real number. Then
 \begin{eqnarray*}
E(|X-Ex(X)|\geq t)\leq \frac{V(X)}{t^2}.
\end{eqnarray*}  
\end{theorem}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $X$ be a Neutrosophic Variable and let $t$ be a positive real number. Then
 \begin{eqnarray*}
E(|X-Ex(X)|\geq t)=E{((X-Ex(X))}^2 \geq t^2)\leq \frac{Ex({(X-Ex(X))}^2)}{t^2}=\frac{V(X)}{t^2}. 􏱫
\end{eqnarray*}  

\end{proof}
\begin{corollary}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $X_n$ be a Neutrosophic Variable in a probability {\tiny Hamiltonian-Cycle-Neighbor} (V_n , E_n ), n \geq 1.$ If $Ex(X_n)\neq 0$ and $V(X_n) << E^2(X_n),$ then
    \begin{eqnarray*}
E(X_n=0)\rightarrow 0~\text{as}~n\rightarrow \infty
\end{eqnarray*}
\end{corollary}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Set $X := X_n$ and $t := |Ex(X_n)|$ in Chebyshev’s Inequality, and observe that $E (X_n = 0) \leq E (|X_n - Ex(X_n)| \geq |Ex(X_n)|)$ because $|X_n - Ex(X_n)| = |Ex(X_n)|$ when $X_n = 0.$
\end{proof}
\begin{theorem}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $G \in \mathcal{G}_{n,1/2}.$ For $0 \leq k \leq n,$ set $f(k) := \text{(n choose k)}􏰇􏰈2^{-\text{(k choose 2)}}$ and let $k^{*}$ be the least value of $k$ for which $f(k)$ is less than one. Then almost surely $\alpha(G)$ takes one of the three values $k^{*}-2,k^{*}-1,k^{*}.$
\end{theorem}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. As in the proof of related Theorem, the result is straightforward.
\end{proof}

\begin{corollary}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $G \in \mathcal{G}_{n,1/2}$ and let $f$ and $k^{*}$ be as defined in previous Theorem. Then either:
   \begin{itemize}
 \item[$(i).$] $f(k^{*}) << 1,$ in which case almost surely $\alpha(G)$ is equal to either  $k^{*}-2$ or  $k^{*}-1$, or
 \item[$(ii).$] $f(k^{*}-1) >> 1,$ in which case almost surely $\alpha(G)$  is equal to either $k^{*}-1$ or $k^{*}.$
\end{itemize}
\end{corollary}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.
\end{proof}
\begin{definition}(Neutrosophic Threshold).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $P$ be a  monotone property of SuperHyperGraphs (one which is preserved when SuperHyperEdges are added). Then a \textbf{Neutrosophic Threshold} for $P$ is a function $f(n)$ such that:
 \begin{itemize}
 \item[$(i).$]  if $p << f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely does not have $P,$
 \item[$(ii).$]  if $p >> f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely has $P.$
\end{itemize}
\end{definition}
\begin{definition}(Neutrosophic Balanced).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $F$ be a fixed Neutrosophic SuperHyperGraph. Then there is a threshold function for the property of containing a copy of $F$ as a Neutrosophic SubSuperHyperGraph is called \textbf{Neutrosophic Balanced}.
\end{definition}
 \begin{theorem}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. Let $F$ be a nonempty balanced Neutrosophic SubSuperHyperGraph with $k$ SuperHyperVertices and $l$ SuperHyperEdges. Then $n^{-k/l}$ is a threshold function for the property of containing $F$ as a Neutrosophic SubSuperHyperGraph.
\end{theorem}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward. $E_1$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
   =\{V_1,V_2,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =z^5+2z^3.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG1.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG1}
\end{figure}
 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
   =\{V_1,V_2,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =z^5+z^3+z.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG2.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG2}
 \end{figure}
  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
   =\{V_1,V_2,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =z^8+z^4+z^3.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG3.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG3}
 \end{figure}
 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG4.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG4}
 \end{figure}
  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
 \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG5.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG5}
 \end{figure}
  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG6.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG6}
 \end{figure}
 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG7.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG7}
 \end{figure}
 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG8.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG8}
 \end{figure}
 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG9.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG9}
 \end{figure}
 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG10.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG10}
 \end{figure}
  \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
  &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG11.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG11}
 \end{figure}
 \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG12.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG12}
 \end{figure}
  \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
  \begin{figure}
 \includegraphics[width=100mm]{136NSHG13.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG13}
 \end{figure}
  \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG14.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG14}
 \end{figure}
  \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG15.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG15}
 \end{figure}
  \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG16.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG16}
 \end{figure}
  \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
  \end{eqnarray*}
   \begin{figure}
 \includegraphics[width=100mm]{136NSHG17.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG17}
 \end{figure}
  \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
  \end{eqnarray*}
   \begin{figure}
 \includegraphics[width=100mm]{136NSHG18.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG18}
 \end{figure}
  \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG19.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG19}
 \end{figure}
  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG20.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG20}
 \end{figure}
 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
     \\&&      
          =\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
  \\&&
 =z^{10}+z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
     \\&&
     =\{V_{E_2}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
    \\&&
  =z^5+2z^3+7z.
 \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG1.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{95NHG1}
 \end{figure}
 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG2.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM1} }
 \label{95NHG2}
 \end{figure}
\end{itemize}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ The all interior Neutrosophic SuperHyperVertices belong to any Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} if for any of them, and any of other corresponded Neutrosophic SuperHyperVertex, some interior Neutrosophic SuperHyperVertices are mutually Neutrosophic  SuperHyperNeighbors with no Neutrosophic exception at all minus  all Neutrosophic SuperHypeNeighbors to any amount of them.
\end{proposition}
\begin{proposition}
Assume a connected non-obvious Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ There's only one  Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior Neutrosophic SuperHyperVertices inside of any given Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} minus all  Neutrosophic SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct Neutrosophic SuperHyperVertices in an  Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}, minus all  Neutrosophic SuperHypeNeighbor to some of them but not all of them.
\end{proposition}
\begin{proposition}
Assume a connected Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ If a Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ Neutrosophic SuperHyperVertices, then the Neutrosophic cardinality of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the Neutrosophic cardinality of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is at least the maximum Neutrosophic number of Neutrosophic SuperHyperVertices of the Neutrosophic SuperHyperEdges with the maximum number of the Neutrosophic SuperHyperEdges. In other words, the maximum number of the Neutrosophic SuperHyperEdges contains the maximum Neutrosophic number of Neutrosophic SuperHyperVertices are renamed to Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} in some cases but the maximum number of the  Neutrosophic SuperHyperEdge with the maximum Neutrosophic number of Neutrosophic SuperHyperVertices, has the Neutrosophic SuperHyperVertices are contained in a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}.
\end{proposition}
\begin{proposition}
 Assume a simple Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then the Neutrosophic number of  type-result-R-{\tiny Hamiltonian-Cycle-Neighbor} has, the least Neutrosophic cardinality, the lower sharp Neutrosophic bound for Neutrosophic cardinality, is the Neutrosophic cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E'},c_{E''},c_{E'''}\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
If there's a Neutrosophic type-result-R-{\tiny Hamiltonian-Cycle-Neighbor} with the least Neutrosophic cardinality, the lower sharp Neutrosophic bound for cardinality.
\end{proposition}

\begin{proposition}
 Assume a connected loopless Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
\begin{eqnarray*}
 &&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_1,E_1,V_2,E_2,V_3,E_3,V_4,E_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=z^4.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_1,V_2,V_3,V_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=z^5.
\end{eqnarray*}
Is a Neutrosophic type-result-{\tiny Hamiltonian-Cycle-Neighbor}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a Neutrosophic type-result-{\tiny Hamiltonian-Cycle-Neighbor} is the cardinality of
\begin{eqnarray*}
 &&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_1,E_1,V_2,E_2,V_3,E_3,V_4,E_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=z^4.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_1,V_2,V_3,V_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=z^5.
\end{eqnarray*}
\end{proposition}

\begin{proof}
Assume a connected loopless Neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} since neither amount of Neutrosophic SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the Neutrosophic number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the Neutrosophic SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices has the eligibilities to propose property such that there's no  Neutrosophic SuperHyperVertex of a Neutrosophic SuperHyperEdge is common and there's an   Neutrosophic SuperHyperEdge for all Neutrosophic SuperHyperVertices but the maximum Neutrosophic cardinality indicates that these Neutrosophic  type-SuperHyperSets couldn't give us the Neutrosophic lower bound in the term of Neutrosophic sharpness. In other words, the Neutrosophic SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the Neutrosophic SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the Neutrosophic SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the Neutrosophic SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless Neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless Neutrosophic SuperHyperClasses of the connected loopless Neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the Neutrosophic SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The Neutrosophic structure of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} decorates the Neutrosophic SuperHyperVertices don't have received any Neutrosophic connections so as this Neutrosophic style implies different versions of Neutrosophic SuperHyperEdges with the maximum Neutrosophic cardinality in the terms of Neutrosophic SuperHyperVertices are spotlight. The lower Neutrosophic bound is to have the maximum Neutrosophic groups of Neutrosophic SuperHyperVertices have perfect Neutrosophic connections inside each of SuperHyperEdges and the outside of this Neutrosophic SuperHyperSet doesn't matter but regarding the connectedness of the used Neutrosophic SuperHyperGraph arising from its Neutrosophic properties taken from the fact that it's simple. If there's no more than one Neutrosophic SuperHyperVertex in the targeted Neutrosophic SuperHyperSet, then there's no Neutrosophic connection. Furthermore, the Neutrosophic existence of one Neutrosophic SuperHyperVertex has no  Neutrosophic effect to talk about the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Since at least two Neutrosophic SuperHyperVertices involve to make a title in the Neutrosophic background of the Neutrosophic SuperHyperGraph. The Neutrosophic SuperHyperGraph is obvious if it has no Neutrosophic SuperHyperEdge but at least two Neutrosophic SuperHyperVertices make the Neutrosophic version of Neutrosophic SuperHyperEdge. Thus in the Neutrosophic setting of non-obvious Neutrosophic SuperHyperGraph, there are at least one Neutrosophic SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as Neutrosophic adjective for the initial Neutrosophic SuperHyperGraph, induces there's no Neutrosophic  appearance of the loop Neutrosophic version of the Neutrosophic SuperHyperEdge and this Neutrosophic SuperHyperGraph is said to be loopless. The Neutrosophic adjective ``loop'' on the basic Neutrosophic framework engages one Neutrosophic SuperHyperVertex but it never happens in this Neutrosophic setting. With these Neutrosophic bases, on a Neutrosophic SuperHyperGraph, there's at least one Neutrosophic SuperHyperEdge thus there's at least a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} has the Neutrosophic cardinality of a Neutrosophic SuperHyperEdge. Thus, a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} has the Neutrosophic cardinality at least a Neutrosophic SuperHyperEdge. Assume a Neutrosophic SuperHyperSet $V\setminus V\setminus \{z\}.$ This Neutrosophic SuperHyperSet isn't a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} since either the Neutrosophic SuperHyperGraph is an obvious Neutrosophic SuperHyperModel thus it never happens since there's no Neutrosophic usage of this Neutrosophic framework and even more there's no Neutrosophic connection inside or the Neutrosophic SuperHyperGraph isn't obvious and as its consequences, there's a Neutrosophic contradiction with the term ``Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}'' since the maximum Neutrosophic cardinality never happens for this Neutrosophic style of the Neutrosophic SuperHyperSet and beyond that there's no Neutrosophic connection inside as mentioned in first Neutrosophic case in the forms of drawback for this selected Neutrosophic SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This Neutrosophic case implies having the Neutrosophic style of on-quasi-triangle Neutrosophic style on the every Neutrosophic elements of this Neutrosophic SuperHyperSet. Precisely, the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices such that some Neutrosophic amount of the Neutrosophic SuperHyperVertices are on-quasi-triangle Neutrosophic style. The Neutrosophic cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the Neutrosophic SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower Neutrosophic bound is up. Thus the minimum Neutrosophic cardinality of the maximum Neutrosophic cardinality ends up the Neutrosophic discussion. The first Neutrosophic term refers to the Neutrosophic setting of the Neutrosophic SuperHyperGraph but this key point is enough since there's a Neutrosophic SuperHyperClass of a Neutrosophic SuperHyperGraph has no on-quasi-triangle Neutrosophic style amid some amount of its Neutrosophic SuperHyperVertices. This Neutrosophic setting of the Neutrosophic SuperHyperModel proposes a Neutrosophic SuperHyperSet has only some amount  Neutrosophic SuperHyperVertices from one Neutrosophic SuperHyperEdge such that there's no Neutrosophic amount of Neutrosophic SuperHyperEdges more than one involving these some amount of these Neutrosophic SuperHyperVertices. The Neutrosophic cardinality of this Neutrosophic SuperHyperSet is the maximum and the Neutrosophic case is occurred in the minimum Neutrosophic situation. To sum them up, the Neutrosophic SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum Neutrosophic cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some Neutrosophic SuperHyperVertices such that there's distinct-covers-order-amount Neutrosophic SuperHyperEdges for amount of Neutrosophic SuperHyperVertices taken from the Neutrosophic SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a Neutrosophic  R-{\tiny Hamiltonian-Cycle-Neighbor} for the Neutrosophic SuperHyperGraph as used Neutrosophic background in the Neutrosophic terms of worst Neutrosophic case and the common theme of the lower Neutrosophic bound occurred in the specific Neutrosophic SuperHyperClasses of the Neutrosophic SuperHyperGraphs which are Neutrosophic free-quasi-triangle.
  \\
Assume a Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  Neutrosophic number of the Neutrosophic SuperHyperVertices. Then every Neutrosophic SuperHyperVertex has at least no Neutrosophic SuperHyperEdge with others in common. Thus those Neutrosophic SuperHyperVertices have the eligibles to be contained in a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Those Neutrosophic SuperHyperVertices are potentially included in a Neutrosophic  style-R-{\tiny Hamiltonian-Cycle-Neighbor}. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the Neutrosophic  SuperHyperVertices of the Neutrosophic SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the Neutrosophic SuperHyperVertices and there's only and only one  Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the Neutrosophic SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} but with slightly differences in the maximum Neutrosophic cardinality amid those Neutrosophic type-SuperHyperSets of the Neutrosophic SuperHyperVertices. Thus the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{Neutrosophic cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the Neutrosophic SuperHyperVertices belong to the Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is a Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all Neutrosophic intended SuperHyperVertices but in a Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ If a Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ Neutrosophic SuperHyperVertices, then the Neutrosophic cardinality of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the Neutrosophic cardinality of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} is at least the maximum Neutrosophic number of Neutrosophic SuperHyperVertices of the Neutrosophic SuperHyperEdges with the maximum number of the Neutrosophic SuperHyperEdges. In other words, the maximum number of the Neutrosophic SuperHyperEdges contains the maximum Neutrosophic number of Neutrosophic SuperHyperVertices are renamed to Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} in some cases but the maximum number of the  Neutrosophic SuperHyperEdge with the maximum Neutrosophic number of Neutrosophic SuperHyperVertices, has the Neutrosophic SuperHyperVertices are contained in a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}.
\\
The obvious SuperHyperGraph has no Neutrosophic SuperHyperEdges. But the non-obvious Neutrosophic SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the Neutrosophic optimal SuperHyperObject. It specially delivers some remarks on the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices such that there's distinct amount of Neutrosophic SuperHyperEdges for distinct amount of Neutrosophic SuperHyperVertices up to all  taken from that Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices but this Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices is either has the maximum Neutrosophic SuperHyperCardinality or it doesn't have  maximum Neutrosophic SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one Neutrosophic SuperHyperEdge containing at least all Neutrosophic SuperHyperVertices. Thus it forms a Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} where the Neutrosophic completion of the Neutrosophic incidence is up in that.  Thus it's, literarily, a Neutrosophic embedded R-{\tiny Hamiltonian-Cycle-Neighbor}. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum Neutrosophic SuperHyperCardinality and they're Neutrosophic SuperHyperOptimal. The less than two distinct types of Neutrosophic SuperHyperVertices are included in the minimum Neutrosophic style of the embedded Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. The interior types of the Neutrosophic SuperHyperVertices are deciders. Since the Neutrosophic number of SuperHyperNeighbors are only  affected by the interior Neutrosophic SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the Neutrosophic SuperHyperSet for any distinct types of Neutrosophic SuperHyperVertices pose the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Thus Neutrosophic exterior SuperHyperVertices could be used only in one Neutrosophic SuperHyperEdge and in Neutrosophic SuperHyperRelation with the interior Neutrosophic SuperHyperVertices in that  Neutrosophic SuperHyperEdge. In the embedded Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, there's the usage of exterior Neutrosophic SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One Neutrosophic SuperHyperVertex has no connection, inside. Thus, the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. The Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} with the exclusion of the exclusion of all  Neutrosophic SuperHyperVertices in one Neutrosophic SuperHyperEdge and with other terms, the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} with the inclusion of all Neutrosophic SuperHyperVertices in one Neutrosophic SuperHyperEdge, is a Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}. To sum them up, in a connected non-obvious Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ There's only one  Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior Neutrosophic SuperHyperVertices inside of any given Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} minus all  Neutrosophic SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique Neutrosophic SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct Neutrosophic SuperHyperVertices in an  Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}, minus all  Neutrosophic SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} has two titles. a Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} and its corresponded quasi-maximum Neutrosophic R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any Neutrosophic number, there's a Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} with that quasi-maximum Neutrosophic SuperHyperCardinality in the terms of the embedded Neutrosophic SuperHyperGraph. If there's an embedded Neutrosophic SuperHyperGraph, then the Neutrosophic quasi-SuperHyperNotions lead us to take the collection of all the Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}s for all Neutrosophic numbers less than its Neutrosophic corresponded maximum number. The essence of the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} ends up but this essence starts up in the terms of the Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}, again and more in the operations of collecting all the Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}s acted on the all possible used formations of the Neutrosophic SuperHyperGraph to achieve one Neutrosophic number. This Neutrosophic number is\\ considered as the equivalence class for all corresponded quasi-R-{\tiny Hamiltonian-Cycle-Neighbor}s. Let $z_{\text{Neutrosophic Number}},S_{\text{Neutrosophic SuperHyperSet}}$ and $G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}$ be a Neutrosophic number, a Neutrosophic SuperHyperSet and a Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}. Then
\begin{eqnarray*}
&&[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=\{S_{\text{Neutrosophic SuperHyperSet}}~|
\\&&~S_{\text{Neutrosophic SuperHyperSet}}=G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\\&&~|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{S_{\text{Neutrosophic SuperHyperSet}}~|
\\&&~S_{\text{Neutrosophic SuperHyperSet}}=G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\\&&~|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{S_{\text{Neutrosophic SuperHyperSet}}~|
\\&&~S_{\text{Neutrosophic SuperHyperSet}}=G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\\&&~|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{S_{\text{Neutrosophic SuperHyperSet}}~|
\\&&~S_{\text{Neutrosophic SuperHyperSet}}=G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\\&&~|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{S_{\text{Neutrosophic SuperHyperSet}}~|
\\&&~S_{\text{Neutrosophic SuperHyperSet}}=G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\\&&~|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{S\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|S_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``Neutrosophic SuperHyperNeighborhood'', could be redefined as the collection of the Neutrosophic SuperHyperVertices such that any amount of its Neutrosophic SuperHyperVertices are incident to a Neutrosophic  SuperHyperEdge. It's, literarily,  another name for ``Neutrosophic  Quasi-{\tiny Hamiltonian-Cycle-Neighbor}'' but, precisely, it's the generalization of  ``Neutrosophic  Quasi-{\tiny Hamiltonian-Cycle-Neighbor}'' since ``Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}'' happens ``Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}'' in a Neutrosophic SuperHyperGraph as initial framework and background but ``Neutrosophic SuperHyperNeighborhood'' may not happens ``Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}'' in a Neutrosophic SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the Neutrosophic SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``Neutrosophic SuperHyperNeighborhood'',  ``Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}'', and  ``Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}'' are up.
\\
Thus, let $z_{\text{Neutrosophic Number}},N_{\text{Neutrosophic SuperHyperNeighborhood}}$ and $G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}$ be a Neutrosophic number, a Neutrosophic SuperHyperNeighborhood and a Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{N_{\text{Neutrosophic SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{N_{\text{Neutrosophic SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{N_{\text{Neutrosophic SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}=
\\&&
\cup_{z_{\text{Neutrosophic Number}}}\{N_{\text{Neutrosophic SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&=z_{\text{Neutrosophic Number}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperNeighborhood}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max_{[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}}z_{\text{Neutrosophic Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}=
\\&&
\{N_{\text{Neutrosophic SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{Neutrosophic Number}}}[z_{\text{Neutrosophic Number}}]_{\text{Neutrosophic Class}}~|~
\\&&
|N_{\text{Neutrosophic SuperHyperSet}}|_{\text{Neutrosophic Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ The all interior Neutrosophic SuperHyperVertices belong to any Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} if for any of them, and any of other corresponded Neutrosophic SuperHyperVertex, some interior Neutrosophic SuperHyperVertices are mutually Neutrosophic  SuperHyperNeighbors with no Neutrosophic exception at all minus  all Neutrosophic SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following Neutrosophic SuperHyperSet  of Neutrosophic  SuperHyperVertices is the simple Neutrosophic type-SuperHyperSet of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The Neutrosophic SuperHyperSet of Neutrosophic SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple Neutrosophic type-SuperHyperSet of the Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. The Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}}} $\mathcal{C}(ESHG)$ for an  Neutrosophic SuperHyperGraph $ESHG:(V,E)$ is a Neutrosophic  type-SuperHyperSet with \\\\\underline{\textbf{the maximum Neutrosophic cardinality}}  of a Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperVertices such that there's no a Neutrosophic  SuperHyperEdge amid some Neutrosophic SuperHyperVertices instead of all given by \underline{\textbf{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}} is related to the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} Neutrosophic SuperHyperVertex \underline{\textbf{inside}} the intended Neutrosophic SuperHyperSet. Thus the non-obvious  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} is up. The obvious simple Neutrosophic type-SuperHyperSet called the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} is a Neutrosophic SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} Neutrosophic SuperHyperVertex. But the Neutrosophic SuperHyperSet of Neutrosophic SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended Neutrosophic SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} \underline{\textbf{is}} up. To sum them up, the Neutrosophic SuperHyperSet of Neutrosophic SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Since the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} $\mathcal{C}(ESHG)$ for an  Neutrosophic SuperHyperGraph $ESHG:(V,E)$ is the Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperVertices such that there's no a Neutrosophic  SuperHyperEdge for some  Neutrosophic SuperHyperVertices instead of all given by that Neutrosophic type-SuperHyperSet  called the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} \underline{\textbf{and}} it's an  Neutrosophic \underline{\textbf{ {\tiny Hamiltonian-Cycle-Neighbor}}}. Since it\underline{\textbf{'s}}   \\\\\underline{\textbf{the maximum Neutrosophic cardinality}} of  a Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperVertices such that there's no a Neutrosophic SuperHyperEdge for some amount Neutrosophic  SuperHyperVertices instead of all given by that Neutrosophic type-SuperHyperSet called the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}. There isn't  only less than two Neutrosophic  SuperHyperVertices \underline{\textbf{inside}} the intended  Neutrosophic SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor},
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the Neutrosophic SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected Neutrosophic  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple Neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``Neutrosophic  R-{\tiny Hamiltonian-Cycle-Neighbor}''}}
\end{center}
 amid those obvious[non-obvious] simple Neutrosophic type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected Neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only a Neutrosophic free-triangle embedded SuperHyperModel and a Neutrosophic on-triangle embedded SuperHyperModel but also it's a Neutrosophic stable embedded SuperHyperModel. But all only non-obvious simple Neutrosophic type-SuperHyperSets of the Neutrosophic  R-{\tiny Hamiltonian-Cycle-Neighbor} amid those obvious simple Neutrosophic type-SuperHyperSets of the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a Neutrosophic  R-{\tiny Hamiltonian-Cycle-Neighbor} is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ The all interior Neutrosophic SuperHyperVertices belong to any Neutrosophic quasi-R-{\tiny Hamiltonian-Cycle-Neighbor} if for any of them, and any of other corresponded Neutrosophic SuperHyperVertex, some interior Neutrosophic SuperHyperVertices are mutually Neutrosophic  SuperHyperNeighbors with no Neutrosophic exception at all minus  all Neutrosophic SuperHypeNeighbors to any amount of them.
\\
Assume a connected Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ Let a Neutrosophic SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some Neutrosophic SuperHyperVertices $r.$ Consider all Neutrosophic numbers of those Neutrosophic SuperHyperVertices from that Neutrosophic SuperHyperEdge excluding excluding more than $r$ distinct Neutrosophic SuperHyperVertices, exclude to any given Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices. Consider there's a Neutrosophic  R-{\tiny Hamiltonian-Cycle-Neighbor} with the least cardinality, the lower sharp Neutrosophic bound for Neutrosophic cardinality. Assume a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ The Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is a Neutrosophic SuperHyperSet $S$ of  the Neutrosophic SuperHyperVertices such that there's a Neutrosophic SuperHyperEdge to have  some Neutrosophic SuperHyperVertices uniquely but it isn't a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Since it doesn't have  \\\\\underline{\textbf{the maximum Neutrosophic cardinality}} of a Neutrosophic SuperHyperSet $S$ of  Neutrosophic SuperHyperVertices such that there's a Neutrosophic SuperHyperEdge to have some SuperHyperVertices uniquely. The Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum Neutrosophic cardinality of a Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperVertices but it isn't a Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor}. Since it \textbf{\underline{doesn't do}} the Neutrosophic procedure such that such that there's a Neutrosophic SuperHyperEdge to have some Neutrosophic  SuperHyperVertices uniquely  [there are at least one Neutrosophic SuperHyperVertex outside  implying there's, sometimes in  the connected Neutrosophic SuperHyperGraph $ESHG:(V,E),$ a Neutrosophic SuperHyperVertex, titled its Neutrosophic SuperHyperNeighbor,  to that Neutrosophic SuperHyperVertex in the Neutrosophic SuperHyperSet $S$ so as $S$ doesn't do ``the Neutrosophic procedure''.]. There's  only \textbf{\underline{one}} Neutrosophic SuperHyperVertex    \textbf{\underline{outside}} the intended Neutrosophic SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of Neutrosophic SuperHyperNeighborhood. Thus the obvious Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor},  $V_{ESHE}$ is up. The obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor},  $V_{ESHE},$ \textbf{\underline{is}} a Neutrosophic SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  Neutrosophic SuperHyperVertices does forms any kind of Neutrosophic pairs are titled   \underline{Neutrosophic SuperHyperNeighbors} in a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Since the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum Neutrosophic SuperHyperCardinality}} of a Neutrosophic SuperHyperSet $S$ of  Neutrosophic SuperHyperVertices  \textbf{\underline{such that}}  there's a Neutrosophic SuperHyperEdge to have some Neutrosophic SuperHyperVertices uniquely. Thus, in a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$ Any Neutrosophic R-{\tiny Hamiltonian-Cycle-Neighbor} only contains all interior Neutrosophic SuperHyperVertices and all exterior Neutrosophic SuperHyperVertices from the unique Neutrosophic SuperHyperEdge where there's any of them has all possible  Neutrosophic SuperHyperNeighbors in and there's all  Neutrosophic SuperHyperNeighborhoods in with no exception minus all  Neutrosophic SuperHypeNeighbors to some of them not all of them  but everything is possible about Neutrosophic SuperHyperNeighborhoods and Neutrosophic SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  {\tiny Hamiltonian-Cycle-Neighbor}, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following Neutrosophic SuperHyperSet  of Neutrosophic  SuperHyperEdges[SuperHyperVertices] is the simple Neutrosophic type-SuperHyperSet of the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}.
 The Neutrosophic SuperHyperSet of Neutrosophic SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple Neutrosophic type-SuperHyperSet of the Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}. The Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}} $\mathcal{C}(ESHG)$ for an  Neutrosophic SuperHyperGraph $ESHG:(V,E)$ is a Neutrosophic  type-SuperHyperSet with \\\\\underline{\textbf{the maximum Neutrosophic cardinality}}  of a Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperEdges[SuperHyperVertices] such that there's no  Neutrosophic SuperHyperVertex of a Neutrosophic SuperHyperEdge is common and there's an   Neutrosophic SuperHyperEdge for all Neutrosophic SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} Neutrosophic SuperHyperVertices \underline{\textbf{inside}} the intended Neutrosophic SuperHyperSet. Thus the non-obvious  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} is up. The obvious simple Neutrosophic type-SuperHyperSet called the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} is a Neutrosophic SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} Neutrosophic SuperHyperVertices. But the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended Neutrosophic SuperHyperSet. Thus the non-obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} \underline{\textbf{is}} up. To sum them up, the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}. Since the Neutrosophic SuperHyperSet of the Neutrosophic SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} $\mathcal{C}(ESHG)$ for an  Neutrosophic SuperHyperGraph $ESHG:(V,E)$ is the Neutrosophic SuperHyperSet $S$ of Neutrosophic SuperHyperVertices such that there's no a Neutrosophic  SuperHyperEdge for some  Neutrosophic SuperHyperVertices given by that Neutrosophic type-SuperHyperSet  called the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} \underline{\textbf{and}} it's an  Neutrosophic \underline{\textbf{ {\tiny Hamiltonian-Cycle-Neighbor}}}. Since it\underline{\textbf{'s}}   \\\\\underline{\textbf{the maximum Neutrosophic cardinality}} of  a Neutrosophic SuperHyperSet $S$ of  Neutrosophic SuperHyperEdges[SuperHyperVertices] such that there's no  Neutrosophic SuperHyperVertex of a Neutrosophic SuperHyperEdge is common and there's an   Neutrosophic SuperHyperEdge for all Neutrosophic SuperHyperVertices. There aren't  only less than three Neutrosophic  SuperHyperVertices \underline{\textbf{inside}} the intended  Neutrosophic SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple Neutrosophic type-SuperHyperSet of the  Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the Neutrosophic SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected Neutrosophic  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple Neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``Neutrosophic  {\tiny Hamiltonian-Cycle-Neighbor}''}}
\end{center}
 amid those obvious[non-obvious] simple Neutrosophic type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Neutrosophic Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Neutrosophic R-Quasi-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected Neutrosophic SuperHyperGraph $ESHG:(V,E).$
 \end{proof}

\section{The Neutrosophic Departures on The Theoretical Results Toward Theoretical Motivations}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
  &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
 \end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{a Neutrosophic SuperHyperPath Associated to the Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Example \eqref{136EXM18a}}
\label{136NSHG18a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
  &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{a Neutrosophic SuperHyperCycle Associated to the Neutrosophic Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM19a}}
\label{136NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
 \end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&CENTER,E_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,CENTER
 \end{eqnarray*}
be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{a Neutrosophic SuperHyperStar Associated to the Neutrosophic Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM20a}}
\label{136NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
  is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward. Then there's no at least one SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{Neutrosophic SuperHyperBipartite Neutrosophic Associated to the Neutrosophic Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Example \eqref{136EXM21a}}
\label{136NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
 is a longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward.  Then there's no at least one SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{a Neutrosophic  SuperHyperMultipartite Associated to the Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Example \eqref{136EXM22a}}
\label{136NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor}}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic {\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor}}}
    \\&&
 =\{CENTER\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-{\tiny Hamiltonian-Cycle-Neighbor} SuperHyperPolynomial}}}
   \\&&
 =z^{a}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E^{*}_1,
 \\&&CENTER,E^{*}_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E^{*}_1,V^{EXTERNAL}_1,
 \\&&E^{*}_2,CENTER
 \end{eqnarray*}
  is a longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The latter is straightforward. Then there's at least one SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} could be applied. The unique embedded SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} proposes some longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG23.png}
\caption{a Neutrosophic SuperHyperWheel Neutrosophic Associated to the Neutrosophic Notions of  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in the Neutrosophic Example \eqref{136EXM23a}}
\label{136NSHG23a}
\end{figure}
            \end{example}
\section{The Surveys of Mathematical Sets On The Results But As The Initial Motivation}
For the SuperHyper{\tiny Hamiltonian-Cycle-Neighbor},  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, and the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, some general results are introduced.
\begin{remark}
 Let remind that the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is ``redefined'' on the positions of the alphabets.
\end{remark}
\begin{corollary}
 Assume Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Then
 \begin{eqnarray*}
&& Neutrosophic ~SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}=\\&&\{the   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive \\&&SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}
\\&&
|_{Neutrosophic cardinality amid those SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.}\}
  \end{eqnarray*}
plus one Neutrosophic SuperHypeNeighbor to one. Where $\sigma_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then the notion of Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} coincide.
\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a conseNeighborive sequence of the SuperHyperVertices is a Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if and only if it's a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a conseNeighborive sequence of the SuperHyperVertices is a strongest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} if and only if it's a longest SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a Neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and reversely.
\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperPath(-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel) on the same identical letter of the alphabet. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and reversely.
\end{corollary}
\begin{corollary}
 Assume a Neutrosophic SuperHyperGraph. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a Neutrosophic SuperHyperGraph. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined.
\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperPath(-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume a Neutrosophic SuperHyperGraph. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a Neutrosophic SuperHyperGraph. Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined.
\end{corollary}
\begin{corollary}
Assume a Neutrosophic SuperHyperPath(-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined if and only if its SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} is well-defined.
\end{corollary}

%

\begin{proposition}
Let $ESHG:(V,E)$  be a Neutrosophic SuperHyperGraph. Then $V$ is
\begin{itemize}
 \item[$(i):$]  the dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  the strong dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  the connected dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  the $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  the strong $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  the connected $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $NTG:(V,E,\sigma,\mu)$  be a Neutrosophic SuperHyperGraph. Then $\emptyset$ is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  the connected defensive SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperGraph. Then an independent SuperHyperSet is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  the connected SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}/SuperHyperPath. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  connected SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperGraph which is a  SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}/SuperHyperPath. Then the number of
\begin{itemize}
 \item[$(i):$] the SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$] the SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  the connected SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  the $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  the strong $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  the connected $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of
\begin{itemize}
 \item[$(i):$] the dual SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$] the dual  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  the dual connected SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  the dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  the strong dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  the connected dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices is a
\begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart is a
\begin{itemize}
 \item[$(i):$]
 SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  connected  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
is one and it's only $S,$ a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices. Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$  be a Neutrosophic SuperHyperGraph. The number of connected component is $|V-S|$ if there's a SuperHyperSet which is a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong 1-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected 1-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$  be a Neutrosophic SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and  the Neutrosophic number is at most $\mathcal{O}_n(ESHG).$
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$  be a Neutrosophic SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the Neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of dual
\begin{itemize}
 \item[$(i):$]
 SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$] strong
  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperGraph which is $\emptyset.$ The number is  $0$ and  the Neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of dual
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $0$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $0$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $0$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
 Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperGraph which is SuperHyperComplete. Then there's no independent SuperHyperSet.
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Neutrosophic SuperHyperGraph which is SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}/SuperHyperPath/SuperHyperWheel. The number is  $\mathcal{O}(ESHG:(V,E))$ and  the Neutrosophic number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$  be a Neutrosophic SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the Neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
 Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ Neutrosophic SuperHyperGraphs which are from one-type SuperHyperClass which the result is obtained for the individuals. Then the results also hold for the SuperHyperFamily $\mathcal{NSHF}:(V,E)$ of these specific SuperHyperClasses of the  Neutrosophic SuperHyperGraphs.
\end{proposition}
%
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Neutrosophic SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then $\forall v\in V\setminus S,~\exists x\in S$ such that
   \begin{itemize}
\item[$(i)$]
$v\in N_s(x);$
\item[$(ii)$]
$vx\in E.$
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$  be a strong Neutrosophic SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then
    \begin{itemize}
\item[$(i)$]
$S$ is SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} set;
\item[$(ii)$]
there's $S\subseteq S'$ such that $|S'|$ is SuperHyperChromatic number.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Neutrosophic SuperHyperGraph. Then
      \begin{itemize}
\item[$(i)$]
$\Gamma\leq\mathcal{O};$
\item[$(ii)$]
$\Gamma_s\leq\mathcal{O}_n.$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Neutrosophic SuperHyperGraph which is connected. Then
      \begin{itemize}
\item[$(i)$]
$\Gamma\leq\mathcal{O}-1;$
\item[$(ii)$]
$\Gamma_s\leq\mathcal{O}_n-\Sigma_{i=1}^{3}\sigma_i(x).$
\end{itemize}
\end{proposition}

\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperPath. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor+1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots,v_{n-1}\}$;
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots,v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots,v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ are only a dual SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperPath. Then
 \begin{itemize}
\item[$(i)$]
the set $S=\{v_2,v_4,\cdots.v_{n}\}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots.v_n\}$ and $\{v_1,v_3,\cdots.v_{n-1}\};$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_n\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots.v_{n}\}$ and $S_2=\{v_1,v_3,\cdots.v_{n-1}\}$ are only dual SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots,v_n\}$ and $\{v_1,v_3,\cdots,v_{n-1}\};$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_n\}}\sigma(s), \Sigma_{s\in S=\{v_1,v_3,\cdots,v_{n-1}\}}\sigma(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots,v_{n}\}$ and $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ are only dual  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is  a  dual  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor+1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots,v_{n-1}\}$;
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots.v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots.v_{n-1}\}$ are only dual SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be SuperHyperStar. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c\}$ is  a dual maximal SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
$\Gamma=1;$
\item[$(iii)$]
$\Gamma_s=\Sigma_{i=1}^3\sigma_i(c);$
\item[$(iv)$]
the SuperHyperSets $S=\{c\}$ and $S\subset S'$ are only dual SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be SuperHyperWheel. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is a dual maximal  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
$\Gamma=|\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}|;$
\item[$(iii)$]
$\Gamma_s=\Sigma_{\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}}\Sigma_{i=1}^3\sigma_i(s);$
\item[$(iv)$]
the SuperHyperSet $\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is only a dual maximal  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperComplete. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor+1;$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}};$
\item[$(iv)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is only a dual  SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperComplete. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor;$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}};$
\item[$(iv)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is only a dual maximal SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of Neutrosophic SuperHyperStars with common Neutrosophic SuperHyperVertex SuperHyperSet. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF};$
\item[$(ii)$]
$\Gamma=m$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\Sigma_{i=1}^m\Sigma_{j=1}^3\sigma_j(c_i)$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{c_1,c_2,\cdots,c_m\}$ and $S\subset S'$ are only dual  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be an $m$-SuperHyperFamily of odd SuperHyperComplete SuperHyperGraphs with common Neutrosophic SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is  a dual maximal SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF};$
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor+1$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}}$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ are only a dual maximal SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of even SuperHyperComplete SuperHyperGraphs with common Neutrosophic SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is  a dual SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF}:(V,E);$
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}}$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ are only dual  maximal SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ be a strong Neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then $S$ is an s-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then $S$ is a dual s-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$  be a strong Neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t+2$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then $S$ is an s-SuperHyperPowerful  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, then $S$ is a dual s-SuperHyperPowerful  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
   Let $ESHG:(V,E)$  be a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{r}{2}\rfloor+1,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{r}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an V-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual V-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{r}{2}\rfloor+1$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{r}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual V-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph which is a  SuperHyperComplete. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph which is SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< 2$ if $ESHG:(V,E))$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Neutrosophic SuperHyperGraph which is SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< 2,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor};
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\end{itemize}
\end{proposition}

            
 \section{Neutrosophic Applications in Cancer's Neutrosophic Recognition}
The cancer is the Neutrosophic disease but the Neutrosophic model is going to figure out what's going on this Neutrosophic phenomenon. The special Neutrosophic case of this Neutrosophic disease is considered and as the consequences of the model, some parameters are used. The cells are under attack of this disease but the moves of the cancer in the special region are the matter of mind. The Neutrosophic recognition of the cancer could help to find some Neutrosophic treatments for this Neutrosophic disease.
\\
In the following, some Neutrosophic steps are Neutrosophic devised on this disease.
\begin{description}
 \item[Step 1. (Neutrosophic Definition)] The Neutrosophic recognition of the cancer in the long-term Neutrosophic function.
  \item[Step 2. (Neutrosophic Issue)] The specific region has been assigned by the Neutrosophic model [it's called Neutrosophic SuperHyperGraph] and the long Neutrosophic cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be Neutrosophic SuperHyperGraph] to have convenient perception on what's happened and what's done.
 \item[Step 3. (Neutrosophic Model)]  
There are some specific Neutrosophic models, which are well-known and they've got the names, and some general Neutrosophic models. The moves and the Neutrosophic traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a Neutrosophic SuperHyperPath(-/SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} or the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} in those Neutrosophic Neutrosophic SuperHyperModels.
  \section{Case 1: The Initial Neutrosophic Steps Toward Neutrosophic SuperHyperBipartite as  Neutrosophic SuperHyperModel}

 \item[Step 4. (Neutrosophic Solution)]
In the Neutrosophic Figure \eqref{136NSHGaa21aa}, the Neutrosophic SuperHyperBipartite is Neutrosophic highlighted and Neutrosophic featured.
\begin{figure}
 \includegraphics[width=100mm]{136NSHG21.png}
 \caption{a Neutrosophic  SuperHyperBipartite Associated to the Notions of   Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}}
 \label{136NSHGaa21aa}
\end{figure}
\\ By using the Neutrosophic Figure \eqref{136NSHGaa21aa} and the Table \eqref{136TBLaa21aa}, the Neutrosophic SuperHyperBipartite is obtained.
\\
The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHGaa21aa},
is the Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperBipartite}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{136TBLaa21aa}
\end{table}
    \section{Case 2: The Increasing Neutrosophic Steps Toward Neutrosophic SuperHyperMultipartite as Neutrosophic SuperHyperModel}

     \item[Step 4. (Neutrosophic Solution)]
In the Neutrosophic Figure \eqref{136NSHGaa22aa}, the Neutrosophic SuperHyperMultipartite is Neutrosophic highlighted and  Neutrosophic featured.
\begin{figure}
 \includegraphics[width=100mm]{136NSHG22.png}
 \caption{a Neutrosophic  SuperHyperMultipartite Associated to the Notions of   Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}}
 \label{136NSHGaa22aa}
\end{figure}
\\
 By using the Neutrosophic Figure \eqref{136NSHGaa22aa} and the Table \eqref{136TBLaa22aa}, the Neutrosophic SuperHyperMultipartite is obtained.
 \\
 The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHGaa22aa}, is the  Neutrosophic SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperMultipartite}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{136TBLaa22aa}
\end{table}
     \end{description}
\section{Wondering Open Problems But As The Directions To Forming The Motivations}
In what follows, some ``problems'' and some ``questions'' are proposed.
\\
The   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} are defined on a real-world application, titled ``Cancer's Recognitions''.
\begin{question}
Which the else SuperHyperModels could be defined based on Cancer's recognitions?
\end{question}
\begin{question}
Are there some SuperHyperNotions related to   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}?
\end{question}
\begin{question}
Are there some Algorithms to be defined on the SuperHyperModels to compute them?
\end{question}
\begin{question}
Which the SuperHyperNotions are related to beyond the   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}?
\end{question}
\begin{problem}
The   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} do a SuperHyperModel for the Cancer's recognitions and they're based on   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, are there else?
\end{problem}
\begin{problem}
Which the fundamental SuperHyperNumbers are related to these SuperHyperNumbers types-results?
\end{problem}
\begin{problem}
What's the independent research based on Cancer's recognitions concerning the multiple types of SuperHyperNotions?
\end{problem}
\section{Conclusion and  Closing Remarks}
In this section, concluding remarks and closing remarks are represented. The drawbacks of this research are illustrated. Some benefits and some advantages of this research are highlighted.
\\
This research uses some approaches to make Neutrosophic SuperHyperGraphs more understandable. In this endeavor, two SuperHyperNotions are defined on the   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. For that sake in the second definition, the main definition of the Neutrosophic SuperHyperGraph is redefined on the position of the alphabets. Based on the new definition for the Neutrosophic SuperHyperGraph, the new SuperHyperNotion, Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, finds the convenient background to implement some results based on that. Some SuperHyperClasses and some Neutrosophic SuperHyperClasses are the cases of this research on the modeling of the regions where are under the attacks of the cancer to recognize this disease as it's mentioned on the title ``Cancer's Recognitions''. To formalize the instances on the SuperHyperNotion,   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}, the new SuperHyperClasses and SuperHyperClasses, are introduced. Some general results are gathered in the section on the   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor} and the Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}. The clarifications, instances and literature reviews have taken the whole way through. In this research, the literature reviews have fulfilled the lines containing the notions and the results. The SuperHyperGraph and Neutrosophic SuperHyperGraph are the SuperHyperModels on the ``Cancer's Recognitions'' and both bases are the background of this research. Sometimes the cancer has been happened on the region, full of cells, groups of cells and embedded styles. In this segment, the SuperHyperModel proposes some SuperHyperNotions based on the connectivities of the moves of the cancer in the longest and strongest styles with the formation of the design and the architecture are formally called ``  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}'' in the themes of jargons and buzzwords. The prefix ``SuperHyper'' refers to the theme of the embedded styles to figure out the background for the SuperHyperNotions.
\begin{table}[ht]
\centering
\caption{An Overlook On This Research And Beyond}
\label{136TBLTBL}
\begin{tabular}[t]{|c|c|}
\hline
\textcolor{black}{Advantages}&\textcolor{black}{Limitations}\\
\hline
\textcolor{black}{1. }\textcolor{red}{Redefining  Neutrosophic SuperHyperGraph} &\textcolor{black}{1. }\textcolor{blue}{General Results} \\  &
\\
 \textcolor{black}{2. }\textcolor{red}{  SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}}& \\ &
\\
\textcolor{black}{3. } \textcolor{red}{Neutrosophic   SuperHyper{\tiny Hamiltonian-Cycle-Neighbor}}  &\textcolor{black}{2. } \textcolor{blue}{Other SuperHyperNumbers}
\\&
\\
\textcolor{black}{4. }\textcolor{red}{Modeling of Cancer's Recognitions}    &  \\&
\\
\textcolor{black}{5. }\textcolor{red}{SuperHyperClasses}    &\textcolor{black}{3. }\textcolor{blue}{SuperHyperFamilies}   \\
\hline
\end{tabular}
\end{table}
In the Table \eqref{136TBLTBL}, benefits and avenues for this research are, figured out, pointed out and spoken out.
      
      \section{
Neutrosophic SuperHyperDuality But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperDuality).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperDuality} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperDuality).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperDuality} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperDuality} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperDuality;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperDuality SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperDuality; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperDuality} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(vi)$]
a \textbf{Neutrosophic R-SuperHyperDuality} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperDuality;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperDuality SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperDuality SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperDuality, Neutrosophic re-SuperHyperDuality, Neutrosophic v-SuperHyperDuality, and Neutrosophic rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperDuality; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperDuality.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperDuality.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
  \\&&
4\times5\times5z^3.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
  \\&&
3\times3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_5,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (2\times1\times2)+(2\times4\times5)z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2)z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (2\times2\times2)z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}=10z.
\end{eqnarray*}

  \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperDuality, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}=4z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=10\times9+10\times6+12\times9+12\times6z^2.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperDuality.

\end{example}

\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperDuality SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperDuality}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperDuality SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}
\\&&=\{E_i\in E_{{P_i}^{ESHG:(V,E)}},~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward. Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperDuality taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}
\\&&=\{E_i\in E_{{P_i}^{ESHG:(V,E)}},~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperDuality taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.  Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality}}=\{E^{*}\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperDuality   SuperHyperPolynomial}}
\\&&=|i~|~E^{*}_i\in E^{*}_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperDuality SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E^{*}_1,
 \\&&V^{EXTERNAL}_2,E^{*}_2,
 \\&&\ldots,
 \\&&E^{*}_{|E^{*}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}},V^{EXTERNAL}_{|E^{*}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}+1}
 \end{eqnarray*}
 is a longest SuperHyperDuality taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E^{*}_z\in E^{*}_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E^{*}_z
\equiv
\\&& \exists! E^{*}_z\in E^{*}_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E^{*}_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward. Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. The unique embedded SuperHyperDuality proposes some longest SuperHyperDuality excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperDuality.
            \end{example}
            \section{
Neutrosophic SuperHyperJoin But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperJoin).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperJoin} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ and $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperJoin} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperJoin} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\not\in E_a;$ and $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperJoin} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\in E_a;$ $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperJoin} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperJoin).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperJoin} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperJoin} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperJoin SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperJoin} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(vi)$]
a \textbf{Neutrosophic R-SuperHyperJoin} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperJoin SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperJoin SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1$ and $E_3$ are some empty Neutrosophic  
 SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperJoin.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_2,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperJoin.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_2,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_2,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
  \\&&
4\times5\times5z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
  \\&&
3\times3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_3,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperJoin, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=\{E_2,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperJoin.
\end{example}

\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperJoin}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperJoin SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward. Then there's no at least one SuperHyperJoin. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperJoin could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperJoin taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperJoin taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.  Then there's no at least one SuperHyperJoin. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperJoin could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
  is a longest SuperHyperJoin taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward. Then there's at least one SuperHyperJoin. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperJoin could be applied. The unique embedded SuperHyperJoin proposes some longest SuperHyperJoin excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperJoin.

            \end{example}
           \section{
Neutrosophic SuperHyperPerfect But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperPerfect).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperPerfect} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperPerfect} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$  and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperPerfect} if $\forall V_i\in V_{ESHG:(V,E)}\setminus V',~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperPerfect} if $\forall V_i\in V_{ESHG:(V,E)}\setminus V',~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperPerfect} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperPerfect).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperPerfect} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperPerfect} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperPerfect;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperPerfect SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperPerfect; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperPerfect} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(vi)$]
a \textbf{Neutrosophic R-SuperHyperPerfect} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperPerfect;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperPerfect SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperPerfect SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperPerfect, Neutrosophic re-SuperHyperPerfect, Neutrosophic v-SuperHyperPerfect, and Neutrosophic rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperPerfect; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1$ and $E_3$ are some empty Neutrosophic  
 SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperPerfect.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperPerfect.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
  \\&&
3\times4\times4z^3.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
  \\&&
3\times2z^2.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_3,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperPerfect, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperPerfect}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperPerfect.
\end{example}

\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward. Then there's no at least one SuperHyperPerfect. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperPerfect could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperPerfect taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperPerfect.

\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperPerfect taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.  Then there's no at least one SuperHyperPerfect. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperPerfect could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperPerfect   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperPerfect SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  is a longest SuperHyperPerfect taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward. Then there's at least one SuperHyperPerfect. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperPerfect could be applied. The unique embedded SuperHyperPerfect proposes some longest SuperHyperPerfect excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperPerfect.
            \end{example}

\section{
Neutrosophic SuperHyperTotal But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperTotal).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperTotal} if $\forall E_i\in E_{ESHG:(V,E)},~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperTotal} if $\forall E_i\in E_{ESHG:(V,E)},~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$  and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperTotal} if $\forall V_i\in V_{ESHG:(V,E)},~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperTotal} if $\forall V_i\in V_{ESHG:(V,E)},~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperTotal} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperTotal).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperTotal} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperTotal} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTotal;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperTotal SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTotal; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperTotal} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(vi)$]
a \textbf{Neutrosophic R-SuperHyperTotal} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTotal;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperTotal SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperTotal SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTotal, Neutrosophic re-SuperHyperTotal, Neutrosophic v-SuperHyperTotal, and Neutrosophic rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTotal; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1$ and $E_3$ are some empty Neutrosophic  
 SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperTotal.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperTotal.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}20z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_{12},E_{13},E_{14}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}10z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=2z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=5z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V^{i\neq 4,5,6}_{i_{i=1}^{8}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_3,E_9,E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=3z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
2z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_2,V_3,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
4\times3z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
2\times4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_{{i+2}_{i=0^{11}}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}=11z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperTotal}}=\{V_{{i+2}_{i=0^{11}}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_6,E_{10}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=9z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
   \\&&
  =|(|V|-1)z^2.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}=\{V_1,V_2\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}}=9z^2.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperTotal, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_3,V_{10},V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=3\times6z^3.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=
         \\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}
 \\&&=z^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperTotal.

\end{example}

\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=
         \\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}
 \\&&=({|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-1)
  \\&&
 z^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_^{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-1}},{V^{EXTERNAL}}_{|E_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}-1}}.
 \end{eqnarray*}
 be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_i,E_j\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal   SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|-1)
\\&&
z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,E_j.
 \end{eqnarray*}
  be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward. Then there's no at least one SuperHyperTotal. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTotal could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperTotal taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperTotal taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.  Then there's no at least one SuperHyperTotal. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTotal could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperTotal.

\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal}}=\{E_i,E_j\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperTotal   SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E^{*}_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{Neutrosophic Cardinality}}}|-1)
\\&&
z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E^{*}_i,CENTER,E_j.
 \end{eqnarray*}
  is a longest SuperHyperTotal taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward. Then there's at least one SuperHyperTotal. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperTotal could be applied. The unique embedded SuperHyperTotal proposes some longest SuperHyperTotal excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperTotal.
            \end{example}
    \section{
Neutrosophic SuperHyperConnected But As The Extensions Excerpt From Dense And Super Forms}        
            
\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperConnected).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperConnected} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ and $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperConnected} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperConnected} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\not\in E_a;$ and $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperConnected} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\in E_a;$ $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperConnected} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperConnected).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperConnected} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperConnected} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperConnected;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperConnected SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperConnected; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperConnected} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(vi)$]
a \textbf{Neutrosophic R-SuperHyperConnected} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperConnected;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperConnected SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperConnected SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperConnected, Neutrosophic re-SuperHyperConnected, Neutrosophic v-SuperHyperConnected, and Neutrosophic rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperConnected; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1$ and $E_3$ are some empty Neutrosophic  
 SuperHyperEdges but $E_2$ is a loop Neutrosophic SuperHyperEdge and $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperConnected.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.  $E_1,E_2$ and $E_3$ are some empty Neutrosophic SuperHyperEdges but $E_4$ is a Neutrosophic SuperHyperEdge. Thus in the terms of Neutrosophic SuperHyperNeighbor, there's only one Neutrosophic SuperHyperEdge, namely, $E_4.$ The Neutrosophic SuperHyperVertex, $V_3$ is Neutrosophic isolated means that there's no Neutrosophic SuperHyperEdge has it as a Neutrosophic endpoint. Thus the Neutrosophic SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Neutrosophic SuperHyperConnected.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_1,E_2,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperConnected}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}20z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperConnected}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_{12},E_{13},E_{14}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}10z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_{{i+1}_{i=11}^{19}},V_{22}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_1,E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=2z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=5z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V^{i\neq 4,5,6}_{i_{i=1}^{8}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_3,E_9,E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=3z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_2,V_3,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
4\times3z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
2\times4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}=\{E_{{i+2}_{i=0^{11}}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}=11z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperConnected}}=\{V_{{i+2}_{i=0^{11}}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}
 =z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}}=10z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperConnected, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_3,V_{10},V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=3\times6z^3.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}=
         \\&&=\{E_i\}_{i=1}^{|{E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}
 \\&&=z^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperConnected}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{{|{E}_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-1},{V^{EXTERNAL}}_{{|{E}_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-1}.
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperConnected.
\end{example}

\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}=
         \\&&=\{E_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}
 \\&&=({|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-1)
  \\&&
 z^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic R-Quasi-SuperHyperConnected}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-1},{V^{EXTERNAL}}_{{|E_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}-1}.
 \end{eqnarray*}
 be a longest path taken from a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_i\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E}_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic R-SuperHyperConnected SuperHyperPolynomial}}=
z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,E_j.
 \end{eqnarray*}
  be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward. Then there's no at least one SuperHyperConnected. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperConnected could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperConnected taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Neutrosophic Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperConnected taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.  Then there's no at least one SuperHyperConnected. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperConnected could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperConnected.

\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected}}=\{E_i\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic SuperHyperConnected SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E^{*}}_{ESHG:(V,E)}|}_{\text{Neutrosophic Cardinality}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Neutrosophic V-SuperHyperConnected SuperHyperPolynomial}}=
z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:{V^{EXTERNAL}}_i,{E^{*}}_i,CENTER,E_j.
 \end{eqnarray*}
  is a longest SuperHyperConnected taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward. Then there's at least one SuperHyperConnected. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperConnected could be applied. The unique embedded SuperHyperConnected proposes some longest SuperHyperConnected excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperConnected.
            \end{example}
\section{Background}
There are some scientific researches covering the topic of this research. In what follows, there are some discussion and literature reviews about them.
\\
The seminal paper and groundbreaking article is titled ``neutrosophic co-degree and neutrosophic degree alongside chromatic numbers in the setting of some classes related to neutrosophic hypergraphs'' in \textbf{Ref.} \cite{HG1} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on general forms without using neutrosophic classes of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Current Trends in Computer Science Research (JCTCSR)” with ISO abbreviation ``J Curr Trends Comp Sci Res'' in volume 1 and issue 1 with pages 06-14. The research article studies deeply with choosing neutrosophic hypergraphs instead of neutrosophic SuperHyperGraph. It's the breakthrough toward independent results based on initial background.
\\
The seminal paper and groundbreaking article is titled ``Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes'' in \textbf{Ref.} \cite{HG2} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on fundamental SuperHyperNumber and using neutrosophic SuperHyperClasses of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Mathematical Techniques and Computational Mathematics(JMTCM)” with ISO abbreviation ``J Math Techniques Comput Math'' in volume 1 and issue 3 with pages 242-263. The research article studies deeply with choosing directly neutrosophic SuperHyperGraph and SuperHyperGraph. It's the breakthrough toward independent results based on initial background and fundamental SuperHyperNumbers.
\\
The seminal paper and groundbreaking article is titled ``Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG3} by Henry Garrett (2023). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on fundamental notions and using vital tools in Cancer’s Treatments. It's published in prestigious and fancy journal is entitled “Journal of Mathematical Techniques and Computational Mathematics(JMTCM)” with ISO abbreviation ``J Math Techniques Comput Math'' in volume 2 and issue 1 with pages 35-47. The research article studies deeply with choosing directly neutrosophic SuperHyperGraph and SuperHyperGraph. It's the breakthrough toward independent results based on initial background and fundamental SuperHyperNumbers.
\\
In some articles are titled ``0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph'' in \textbf{Ref.} \cite{HG4} by Henry Garrett (2022),  ``0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs'' in \textbf{Ref.} \cite{HG5} by Henry Garrett (2022),  ``Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG6} by Henry Garrett (2022), ``Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition'' in \textbf{Ref.} \cite{HG7} by Henry Garrett (2022), ``Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG8} by Henry Garrett (2022), ``The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph'' in \textbf{Ref.} \cite{HG9} by Henry Garrett (2022), ``Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG10} by Henry Garrett (2022), ``Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG11} by Henry Garrett (2022), ``Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG12} by Henry Garrett (2022), ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG13} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG14} by Henry Garrett (2022),  ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG15} by Henry Garrett (2022),  ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs
'' in \textbf{Ref.} \cite{HG16} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG12} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG17} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG18} by Henry Garrett (2022),``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances'' in \textbf{Ref.} \cite{HG19} by Henry Garrett (2022), ``(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses'' in \textbf{Ref.} \cite{HG20} by Henry Garrett (2022), ``SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions'' in \textbf{Ref.} \cite{HG21} by Henry Garrett (2022), ``Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG22} by Henry Garrett (2022), ``SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses'' in \textbf{Ref.} \cite{HG23} by Henry Garrett (2022), ``SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG24} by Henry Garrett (2023), ``The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG25} by Henry Garrett (2023), ``Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG26} by Henry Garrett (2023), ``Indeterminacy On The All Possible Connections of Cells In Front of Cancer's Attacks In The Terms of Neutrosophic Failed SuperHyperClique on Cancer's Recognition called Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG27} by Henry Garrett (2023), ``Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG28} by Henry Garrett (2023), ``Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique'' in \textbf{Ref.} \cite{HG29} by Henry Garrett (2023), ``Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG30} by Henry Garrett (2023), ``Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG31} by Henry Garrett (2023), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG32} by Henry Garrett (2023), ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG33} by Henry Garrett (2023), ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG34} by Henry Garrett (2022),  ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG35} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG36} by Henry Garrett (2022), ``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph'' in \textbf{Ref.} \cite{HG37} by Henry Garrett (2022), ``Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)''  in \textbf{Ref.} \cite{HG38} by Henry Garrett (2022), and \cite{HG4,HG5,HG6,HG7,HG8,HG9,HG10,HG11,HG12,HG13,HG14,HG15,HG16,HG17,HG18,HG19,HG20,HG21,HG22,HG23,HG24,HG25,HG26,HG27,HG28,HG29,HG30,HG31,HG32,HG33,HG34,HG35,HG36,HG37,HG38,HG94,HG942,HG95,HG952,HG96,HG962,HG97,HG972,HG98,HG982,HG106,HG107,HG111,HG112,HG115,HG116,HG120,HG121,HG122,HG123,HG124,HG125,HG126,HG127,HG128,HG129,HG130,HG131,HG132,HG134,HG135,HG136,HG137,HG138,HG139,HG140,HG141,HG142,HG143,HG144,HG145,HG146,HG147,HG148,HG149,HG150,HG151,HG152,HG153,HG154,HG155,HG156,HG157,HG158,HG159,HG160,HG161,HG162,HG163,HG164,HG165,HG166,HG167,HG168,HG169,HG170,HG171,HG172,HG173,HG174,HG175,HG176,HG177,HG178,HG179,HG180,HG181,HG182,HG183,HG184,HG185,HG186,HG187,HG188,HG189,HG190,HG191,HG192,HG193,HG194,HG195,HG196,HG197,HG198,HG199,HG200,HG201,HG202,HG203,HG204,HG205,HG206,HG207,HG208,HG209,HG210,HG211,HG212,HG213,HG214,HG215,HG216,HG217,HG218,HG219,HG220,HG221,HG222,HG223,HG224,HG225,HG226,HG228}, there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph alongside scientific research books at \cite{HG60b,HG61b,HG62b,HG63b,HG64b,HG65b,HG66b,HG67b,HG68b,HG69b,HG70b,HG71b,HG72b,HG73b,HG74b,HG75b,HG76b,HG77b,HG78b,HG79b,HG80b,HG81b,HG82b,HG83b,HG84b,HG85b,HG86b,HG87b,HG88b,HG89b,HG90b,HG91b,HG92b,HG93b,HG94b,HG95b,HG96b,HG97b,HG98b,HG99b,HG100b,HG101b,HG102b,HG103b,HG104b,HG105b,HG106b,HG107b,HG108b,HG109b,HG110b,HG111b,HG112b,HG113b,HG114b,HG115b,HG116b,HG117b,HG118b,HG119b,HG120b,HG121b,HG122b,HG123b,HG124b,HG125b,HG126b,HG127b,HG128b,HG129b,HG130b,HG131b,HG132b,HG133b,HG134b,HG135b,HG136b,HG137b,HG138b,HG139b,HG140b,HG141b,HG142b,HG143b,HG144b,HG145b,HG146b,HG147b,HG148b,HG149b,HG150b,HG151b,HG152b,HG153b,HG154b}. Two popular scientific research books in Scribd in the terms of high readers, 4190 and 5189 respectively,  on neutrosophic science is on \cite{HG32b,HG44b}.  
\\
Some scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG71b} by Henry Garrett (2023) which is indexed by Google Scholar and has more than 4190 readers in Scribd. It's titled ``Beyond Neutrosophic Graphs'' and published by Dr. Henry Garrett. This research book covers different types of notions and settings in neutrosophic graph theory and neutrosophic SuperHyperGraph theory.
\\
Also, some scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG70b} by Henry Garrett (2023) which is indexed by Google Scholar and has more than 5189 readers in Scribd. It's titled ``Neutrosophic Duality'' and published by Dr. Henry Garrett. This research book presents different types of notions SuperHyperResolving and SuperHyperDominating in the setting of duality in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. This research book has scrutiny on the complement of the intended set and the intended set, simultaneously. It's smart to consider a set but acting on its complement that what's done in this research book which is popular in the terms of high readers in Scribd.
\\
See the seminal scientific researches \cite{HG1,HG2,HG3}. The formalization of the notions on the framework of notions in SuperHyperGraphs, Neutrosophic notions in SuperHyperGraphs theory, and (Neutrosophic) SuperHyperGraphs theory at \cite{HG4,HG5,HG6,HG7,HG8,HG9,HG10,HG11,HG12,HG13,HG14,HG15,HG16,HG17,HG18,HG19,HG20,HG21,HG22,HG23,HG24,HG25,HG26,HG27,HG28,HG29,HG30,HG31,HG32,HG33,HG34,HG35,HG36,HG37,HG38,HG94,HG942,HG95,HG952,HG96,HG962,HG97,HG972,HG98,HG982,HG106,HG107,HG111,HG112,HG115,HG116,HG120,HG121,HG122,HG123,HG124,HG125,HG126,HG127,HG128,HG129,HG130,HG131,HG132,HG134,HG135,HG136,HG137,HG138,HG139,HG140,HG141,HG142,HG143,HG144,HG145,HG146,HG147,HG148,HG149,HG150,HG151,HG152,HG153,HG154,HG155,HG156,HG157,HG158,HG159,HG160,HG161,HG162,HG163,HG164,HG165,HG166,HG167,HG168,HG169,HG170,HG171,HG172,HG173,HG174,HG175,HG176,HG177,HG178,HG179,HG180,HG181,HG182,HG183,HG184,HG185,HG186,HG187,HG188,HG189,HG190,HG191,HG192,HG193,HG194,HG195,HG196,HG197,HG198,HG199,HG200,HG201,HG202,HG203,HG204,HG205,HG206,HG207,HG208,HG209,HG210,HG211,HG212,HG213,HG214,HG215,HG216,HG217,HG218,HG219,HG220,HG221,HG222,HG223,HG224,HG225,HG226,HG228} alongside scientific research books at \cite{HG60b,HG61b,HG62b,HG63b,HG64b,HG65b,HG66b,HG67b,HG68b,HG69b,HG70b,HG71b,HG72b,HG73b,HG74b,HG75b,HG76b,HG77b,HG78b,HG79b,HG80b,HG81b,HG82b,HG83b,HG84b,HG85b,HG86b,HG87b,HG88b,HG89b,HG90b,HG91b,HG92b,HG93b,HG94b,HG95b,HG96b,HG97b,HG98b,HG99b,HG100b,HG101b,HG102b,HG103b,HG104b,HG105b,HG106b,HG107b,HG108b,HG109b,HG110b,HG111b,HG112b,HG113b,HG114b,HG115b,HG116b,HG117b,HG118b,HG119b,HG120b,HG121b,HG122b,HG123b,HG124b,HG125b,HG126b,HG127b,HG128b,HG129b,HG130b,HG131b,HG132b,HG133b,HG134b,HG135b,HG136b,HG137b,HG138b,HG139b,HG140b,HG141b,HG142b,HG143b,HG144b,HG145b,HG146b,HG147b,HG148b,HG149b,HG150b,HG151b,HG152b,HG153b,HG154b}. Two popular scientific research books in Scribd in the terms of high readers, 4190 and 5189 respectively,  on neutrosophic science is on \cite{HG32b,HG44b}.

\begin{thebibliography}{595}

\bibitem{HG1} Henry Garrett, ``\textit{Neutrosophic Co-degree and Neutrosophic Degree alongside Chromatic Numbers in the Setting of Some Classes Related to Neutrosophic Hypergraphs}'', J Curr Trends Comp Sci Res 1(1) (2022) 06-14.

\bibitem{HG2} Henry Garrett, “Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes”, J Math Techniques Comput Math 1(3) (2022) 242-263. (doi: 10.33140/JMTCM.01.03.09)

\bibitem{HG3} Henry Garrett, “Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments”, J Math Techniques Comput Math 2(1) (2023) 35-47. (https://www.opastpublishers.com/open-access-articles/some-super-hyper-degrees-and-cosuper-hyper-degrees-on-neutrosophic-super-hyper-graphs-and-super-hyper-graphs-alongside-a.pdf)

\bibitem{HG216} Henry Garrett, “A Research on Cancers Recognition and Neutrosophic Super Hypergraph by Eulerian Super Hyper Cycles and Hamiltonian Sets as Hyper Covering Versus Super separations”, J Math Techniques Comput Math 2(3) (2023) 136-148. (https://www.opastpublishers.com/open-access-articles/a-research-on-cancers-recognition-and-neutrosophic-super-hypergraph-by-eulerian-super-hyper-cycles-and-hamiltonian-sets-.pdf)

\bibitem{HG4}
Garrett, Henry. ``\textit{0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph.}'' CERN European Organization for Nuclear Research - Zenodo, Nov. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.5281/zenodo.6319942.
https://oa.mg/work/10.5281/zenodo.6319942

\bibitem{HG5}
Garrett, Henry. ``\textit{0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs.}'' CERN European Organization for Nuclear Research - Zenodo, Feb. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.13140/rg.2.2.35241.26724.
https://oa.mg/work/10.13140/rg.2.2.35241.26724

\bibitem{HG6}
Henry Garrett, ``\textit{Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010308 (doi: 10.20944/preprints202301.0308.v1).

\bibitem{HG7}
Henry Garrett, ``\textit{Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition}'', Preprints 2023, 2023010282 (doi: 10.20944/preprints202301.0282.v1).

\bibitem{HG8}
Henry Garrett, ``\textit{Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010267 (doi: 10.20944/preprints202301.0267.v1).

\bibitem{HG9}
Henry Garrett, ``\textit{The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Preprints 2023, 2023010265 (doi: 10.20944/preprints202301.0265.v1).

\bibitem{HG10}
Henry Garrett, ``\textit{Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010262,(doi: 10.20944/preprints202301.0262.v1).

\bibitem{HG11}
Henry Garrett, ``\textit{Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010240 (doi: 10.20944/preprints202301.0240.v1).

\bibitem{HG12}
Henry Garrett, ``\textit{Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010224, (doi: 10.20944/preprints202301.0224.v1).

\bibitem{HG13}
Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG14}
Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).

\bibitem{HG15}
Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', Preprints 2023, 2023010044

\bibitem{HG16}
Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010043 (doi: 10.20944/preprints202301.0043.v1).

\bibitem{HG17}  Henry Garrett, \textit{``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs''}, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG18}  Henry Garrett, \textit{``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints''}, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).
\bibitem{HG19}  Henry Garrett, \textit{``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances''}, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).
 \bibitem{HG20} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses}'', Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1).
\bibitem{HG21} Henry Garrett, ``\textit{SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions}'', Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1).
\bibitem{HG22} Henry Garrett, ``\textit{Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments}'', Preprints 2022, 2022120324 (doi: 10.20944/preprints202212.0324.v1).
\bibitem{HG23} Henry Garrett, ``\textit{SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses}'', Preprints 2022, 2022110576 (doi: 10.20944/preprints202211.0576.v1).
\bibitem{HG228}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Decomposition As Hyper Decompress On Super Decompensation”, Zenodo 2023, (doi: 10.5281/zenodo.7793372).
\bibitem{HG226}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Cut As Hyper Hamper On Super Hammy”, Zenodo 2023, (doi: 10.5281/zenodo.7791952).
\bibitem{HG225}
Henry Garrett, “New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7791982).
\bibitem{HG224}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Neighbor As Hyper Nebbish On Super Nebulous”, Zenodo 2023, (doi: 10.5281/zenodo.7790026).
\bibitem{HG223}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Hamiltonian-Neighbor In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7790052).
\bibitem{HG222}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Decomposition As Hyper Decompress On Super Decompensation”, Zenodo 2023, (doi: 10.5281/zenodo.7787066).
\bibitem{HG221}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Hamiltonian-Decomposition In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7787094).
\bibitem{HG220}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cut As Hyper Hamper On Super Hammy”, Zenodo 2023, (doi: 10.5281/zenodo.7781476).
\bibitem{HG219}
Henry Garrett, “New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7783082).
\bibitem{HG218}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Trace-Neighbor As Hyper Nebbish On Super Nebulous”, Zenodo 2023, (doi: 10.5281/zenodo.7777857).
\bibitem{HG217}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Trace-Neighbor In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7779286).
\bibitem{HG215}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Trace-Decomposition As Hyper Decompress On Super Decompensation”, Zenodo 2023, (doi: 10.5281/zenodo.7771831).
\bibitem{HG214}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Trace-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7772468).
\bibitem{HG213}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Trace-Cut As Hyper Nebbish On Super Nebulous”, ResearchGate 2023, (doi: 10.13140/RG.2.2.20913.25446).
\bibitem{HG212}
Henry Garrett, “New Ideas On Super Tract By Hyper Track Of Trace-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7764916).
\bibitem{HG211}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Neighbor As Hyper Nebbish On Super Nebulous”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11770.98247).
\bibitem{HG210}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Edge-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.12400.12808).
\bibitem{HG209}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Decomposition As Hyper Decompress On Super Decompensation”, ResearchGate 2023, (doi: 10.13140/RG.2.2.22545.10089).
\bibitem{HG208}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Edge-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29544.34564).
\bibitem{HG207}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Cut As Hyper Edify On Super Eddy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11377.76644).
\bibitem{HG206}
Henry Garrett, “New Ideas On Super Eddy By Hyper Edify Of Edge-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23750.96329).
\bibitem{HG205}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Neighbor As Hyper Nebbish On Super Nebulous”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31366.24641).
\bibitem{HG204}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Vertex-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34721.68960).
\bibitem{HG203}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Decomposition As Hyper Decompress On Super Decompensation”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30212.81289).
\bibitem{HG202}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Vertex-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.18468.76169).
\bibitem{HG201}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Cut As Hyper Vertu On Super Vertigo”, ResearchGate 2023, (doi: 10.13140/RG.2.2.24288.35842).
\bibitem{HG200}
Henry Garrett, “New Ideas On Super Vertigo By Hyper Vertu Of Vertex-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32467.25124).
\bibitem{HG199}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Stable-Neighbor As Hyper Nebbish On Super Nebulous”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31025.45925).
\bibitem{HG198}
Henry Garrett, “New Ideas On Super Nebulizer By Hyper Nub Of Stable-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17184.25602).
\bibitem{HG197}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Stable-Decompositions As Hyper Stain On Super Stagy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23423.28327).
\bibitem{HG196}
Henry Garrett, “New Ideas On Super Stale By Hyper Stalk Of Stable-Decompositions In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.28456.44805).
\bibitem{HG195}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Stable-Cut As Hyper Stain On Super Stagy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23525.68320).
\bibitem{HG194}
Henry Garrett, “New Ideas On Super Stale By Hyper Stalk Of Stable-Cut In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.20170.24000).
\bibitem{HG193}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Neighbors As Hyper Nebbish On Super Nebulous”, ResearchGate 2023, (doi:  10.13140/RG.2.2.36475.59683).
\bibitem{HG192}
Henry Garrett, “New Ideas On Super Nebulizer By Hyper Nub Of Clique-Neighbors In Cancer's
Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi:  10.13140/RG.2.2.29764.71046).
\bibitem{HG191}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Decompositions As Hyper Decompile On Super Decommission”, ResearchGate 2023, (doi:  10.13140/RG.2.2.18780.87683).
\bibitem{HG190}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Clique-
Decompositions In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.27169.48487).
\bibitem{HG189}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Cut As Hyper Click On Super Cliche”, ResearchGate 2023, (doi: 10.13140/RG.2.2.26134.01603).
\bibitem{HG188}
Henry Garrett, “New Ideas On Super Cliff By Hyper Cling Of Clique-Cut In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.27392.30721).
\bibitem{HG187}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Space As Hyper Spin On Super Spacy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33028.40321).
\bibitem{HG186}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By List-
Coloring As Hyper List On Super Lisle”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21389.20966).
\bibitem{HG185}
Henry Garrett, “New Ideas On Super Lith By Hyper Lite Of List-Coloring In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.16356.04489).
\bibitem{HG184}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Space As Hyper Sparse On Super Spark”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21756.21129).
\bibitem{HG183}
Henry Garrett, “New Ideas On Super Solidarity By Hyper Soul Of Space In Cancer's Recognition With (Extreme) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30983.68009).
\bibitem{HG182}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Connectivity As Hyper Disclosure On Super Closure”, ResearchGate 2023, (doi: 10.13140/RG.2.2.28552.29445).
\bibitem{HG181}
Henry Garrett, “New Ideas On Super Uniform By Hyper Deformation Of Edge-Connectivity In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.10936.21761).
\bibitem{HG180}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Connectivity As Hyper Leak On Super Structure”, ResearchGate 2023, (doi: 10.13140/RG.2.2.35105.89447).
\bibitem{HG179}
Henry Garrett, “New Ideas On Super System By Hyper Explosions Of Vertex-Connectivity In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30072.72960).
\bibitem{HG178}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Tree-Decomposition As Hyper Forward On Super Returns”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31147.52003).
\bibitem{HG177}
Henry Garrett, “New Ideas On Super Nodes By Hyper Moves Of Tree-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32825.24163).
\bibitem{HG176}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Chord As Hyper Excellence On Super Excess”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13059.58401).
\bibitem{HG175}
Henry Garrett, “New Ideas On Super Gap By Hyper Navigations Of Chord In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11172.14720).
\bibitem{HG174}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyper(i,j)-Domination As Hyper Controller On Super Waves”, ResearchGate 2023, (doi: 10.13140/RG.2.2.22011.80165).
\bibitem{HG173}
Henry Garrett, “New Ideas On Super Coincidence By Hyper Routes Of SuperHyper(i,j)-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30819.84003).
\bibitem{HG172}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperEdge-Domination As Hyper Reversion On Super Indirection”, ResearchGate 2023, (doi: 10.13140/RG.2.2.10493.84962).
\bibitem{HG171}
Henry Garrett, “New Ideas On Super Obstacles By Hyper Model Of SuperHyperEdge-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13849.29280).
\bibitem{HG170}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperK-Domination As Hyper k-Actions On Super Patterns”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19944.14086).
\bibitem{HG169}
Henry Garrett, “New Ideas On Super Harmony By Hyper k-Function Of SuperHyperK-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23299.58404).
\bibitem{HG168}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperK-Number As Hyper k-Partition On Super Layers”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33103.76968).
\bibitem{HG167}
Henry Garrett, “New Ideas On Super Gradient By Hyper k-Class Of SuperHyperK-Number In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23037.44003).
\bibitem{HG166}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperOrder As Hyper Enumerations On Super Landmarks”, ResearchGate 2023, (doi: 10.13140/RG.2.2.35646.56641).
\bibitem{HG165}
Henry Garrett, “New Ideas On Super Analogous By Hyper Visions Of SuperHyperOrder In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.18030.48967).
\bibitem{HG164}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperColoring As Hyper Categories On Super Neighbors”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13973.81121).

\bibitem{HG163}
Henry Garrett, “New Ideas On Super Relations By Hyper Identifications Of SuperHyperColoring In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34106.47047).

\bibitem{HG162} Henry Garrett, “New Ideas On Super Contradiction By Hyper Detection of SuperHyperDefensive In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13397.09446).

\bibitem{HG161}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperDimension As Hyper Distinguishing On Super Distances”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31956.88961).

\bibitem{HG160} Henry Garrett, “New Ideas On Super Locations By Hyper Differing Of SuperHyperDimension In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.15179.67361).

\bibitem{HG159}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperDominating As Hyper Closing On Super Messy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21510.45125).
\bibitem{HG158}
Henry Garrett, “New Ideas On Super Missing By Hyper Searching Of SuperHyperDominating In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13121.84321).
\bibitem{HG157}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperConnected As Hyper Group On Super Surge”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11758.69441).
\bibitem{HG156}
Henry Garrett, “New Ideas On Super Outbreak By Hyper Collections Of SuperHyperConnected In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31891.35367).
\bibitem{HG155}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperTotal As Hyper Covering On Super Infections”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19360.87048).
\bibitem{HG154}
Henry Garrett, “New Ideas On Super Extremism By Hyper Treatments Of SuperHyperTotal In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32363.21286).
\bibitem{HG153}
  Henry Garrett, “New Ideas On Super Isolation By Hyper Perfectness Of SuperHyperPerfect In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23266.81602).
\bibitem{HG152}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperPerfect As Hyper Idealism On Super Vacancy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19911.37285).
\bibitem{HG151}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperJoin As Hyper Separations On Super Sorts”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11050.90569).
\bibitem{HG150}
Henry Garrett, “New Ideas On Super connections By Hyper disconnections Of SuperHyperJoin In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17761.79206).
\bibitem{HG149}
Henry Garrett, “New Ideas On Super Mixed-Devastations By Hyper Decisions Of SuperHyperDuality In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34953.52320).
\bibitem{HG148}
Henry Garrett, “New Ideas In Cancer’s Recognition As (Neutrosophic) SuperHyperGraph By SuperHyperDuality As Hyper Imaginations On Super Mixed-Illustrations”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33275.80161).
\bibitem{HG147}
Henry Garrett, “New Ideas In Cancer’s Recognition As (Neutrosophic) SuperHyperGraph By Path SuperHyperColoring As Hyper Correction On Super Faults”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30182.50241).
\bibitem{HG146}
Henry Garrett, “New Ideas On Super Reflections By Hyper Rotations Of Path SuperHyperColoring In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33459.30243).
\bibitem{HG145}
Henry Garrett, “New Ideas As Hyper Deformations On Super Chains In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph By SuperHyperDensity”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13444.60806).
\bibitem{HG144}
Henry Garrett, “New Ideas As Hyper Ignorance By SuperHyperDensity On Super Resistances In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi:10.13140/RG.2.2.16800.05123).
\bibitem{HG143}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-VI”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29913.80482).
\bibitem{HG142}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-V”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33269.24809).                
\bibitem{HG141}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-IV”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34946.96960).
\bibitem{HG140}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-III”, ResearchGate 2023, (doi: 10.13140/RG.2.2.14814.31040).
\bibitem{HG139}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-II”, ResearchGate 2023, (doi: 10.13140/RG.2.2.15653.17125).
\bibitem{HG138}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-I”, ResearchGate 2023, (doi: 10.13140/RG.2.2.25719.50089).
\bibitem{HG137}
Henry Garrett, “New Ideas On Super Disruptions In Cancer’s Extreme Recognition As Neutrosophic SuperHyperGraph By Hyper Plans Called SuperHyperConnectivities”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29441.94562).
\bibitem{HG136}
Henry Garrett, “Cancer’s Neutrosophic Recognition As Neutrosophic SuperHyperGraph By SuperHyperConnectivities As Hyper Diagnosis On Super Mechanism”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17252.24968).
\bibitem{HG135}
  Henry Garrett,“Cancer’s Recognition and (Neutrosophic) SuperHyperGraph By the Criteria of Eulerian and Hamiltonian Type-Sets As Hyper Modified Cycles On Super Mess”, ResearchGate 2023,(doi: 10.13140/RG.2.2.16652.59525).
\bibitem{HG134}
Henry Garrett,“Eulerian and Hamiltonian In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph On Super Interactions By Hyper Extensions of Cycles”, ResearchGate 2023,(doi: 10.13140/RG.2.2.34583.24485).

\bibitem{HG132} Henry Garrett,“SuperHyperGirth Type-Results on extreme SuperHyperGirth theory and (Neutrosophic) SuperHyperGraphs Toward Cancer’s extreme Recognition”, Preprints 2023, 2023010396 (doi: 10.20944/preprints202301.0396.v1).

\bibitem{HG131} Henry Garrett,“Neutrosophic SuperHyperGraphs Warns Hyper Landmark of neutrosophic SuperHyperGirth In Super Type-Versions of Cancer’s neutrosophic Recognition”, Preprints 2023, 2023010395 (doi: 10.20944/preprints202301.0395.v1).

\bibitem{HG130} Henry Garrett,“The Constructions of (Neutrosophic) SuperHyperGraphs on the Cancer’s Recognition in The Confrontation With Super Attacks In Hyper Situations By Implementing (Neutrosophic) 1-Failed SuperHyperForcing in The Technical Approaches to Neutralize SuperHyperViews”, ResearchGate 2023, (doi: 10.13140/RG.2.2.26240.51204).

\bibitem{HG129} Henry Garrett,“(Neutrosophic) 1-Failed SuperHyperForcing As the Entrepreneurs on Cancer’s Recognitions To Fail Forcing Style As the Super Classes With Hyper Effects In The Background of the Framework is So-Called (Neutrosophic) SuperHyperGraphs”, ResearchGate 2023, (doi: 10.13140/RG.2.2.12818.73925).

\bibitem{HG128} Henry Garrett,“Super Actions On The Types of Hyper Levels In The Sensible Neutrosophic SuperHyperGirth On Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023,(doi: 10.13140/RG.2.2.26836.88960).

\bibitem{HG127} Henry Garrett,“SuperHyperGirth Approaches on the Super Challenges on the Cancer’s Recognition In the Hyper Model of (Neutrosophic) SuperHyperGraph”, ResearchGate 2023,(doi: 10.13140/RG.2.2.36745.93289).

\bibitem{HG126}  Henry Garrett,“Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010308 (doi: 10.20944/preprints202301.0308.v1).

\bibitem{HG125} Henry Garrett,“Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition”, Preprints 2023, 2023010282 (doi: 10.20944/preprints202301.0282.v1).

\bibitem{HG124}  Henry Garrett,“Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs”, Preprints 2023, 2023010267 (doi: 10.20944/preprints202301.0267.v1).).

\bibitem{HG123}  Henry Garrett, “The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph”, Preprints 2023, 2023010265 (doi: 10.20944/preprints202301.0265.v1).

\bibitem{HG122} Henry Garrett,“Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010262,(doi: 10.20944/preprints202301.0262.v1).

\bibitem{HG121}  Henry Garrett, “Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs”, Preprints 2023, 2023010240 (doi: 10.20944/preprints202301.0240.v1).

\bibitem{HG120} Henry Garrett, “Extremism of the Attacked Body Under the Cancer’s Circumstances Where Cancer’s Recognition Titled (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010224, (doi: 10.20944/preprints202301.0224.v1).

\bibitem{HG24} Henry Garrett,``\textit{SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs}'', ResearchGate 2023,(doi: 10.13140/RG.2.2.35061.65767).

\bibitem{HG25} Henry Garrett,``\textit{The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.18494.15680).

\bibitem{HG26} Henry Garrett,``\textit{Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.32530.73922).

\bibitem{HG27} Henry Garrett,``\textit{Indeterminacy On The All Possible Connections of Cells In Front of Cancer's Attacks In The Terms of Neutrosophic Failed SuperHyperClique on Cancer's Recognition called Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.15897.70243).

\bibitem{HG116}  Henry Garrett,“Extreme Failed SuperHyperClique Decides the Failures on the Cancer’s Recognition in the Perfect Connections of Cancer’s Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32530.73922).

\bibitem{HG115}  Henry Garrett, “(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG28} Henry Garrett,``\textit{Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.30092.80004).

\bibitem{HG29} Henry Garrett,``\textit{Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.23172.19849).

\bibitem{HG112}  Henry Garrett, “Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG111} Henry Garrett, “Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints”, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).

\bibitem{HG30} Henry Garrett,``\textit{Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.17385.36968).

\bibitem{HG107} Henry Garrett, “Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond”, Preprints 2023, 2023010044

\bibitem{HG106} Henry Garrett, “(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010043 (doi: 10.20944/preprints202301.0043.v1).

\bibitem{HG31} Henry Garrett, ``\textit{Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.28945.92007).

\bibitem{HG32} Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.11447.80803).

\bibitem{HG33} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.35774.77123).

\bibitem{HG34} Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.36141.77287).

\bibitem{HG35} Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.29430.88642).

\bibitem{HG36} Henry Garrett, ``\textit{Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.11369.16487).

\bibitem{HG982}  Henry Garrett, “(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances”, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).

\bibitem{HG98}  Henry Garrett, “(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances”, ResearchGate 2022, (doi: 10.13140/RG.2.2.19380.94084).

\bibitem{HG972}  Henry Garrett, “(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses”, Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1).

\bibitem{HG97}  Henry Garrett, “(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses”, ResearchGate 2022, (doi: 10.13140/RG.2.2.14426.41923).

\bibitem{HG962} Henry Garrett, “SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions”, Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1).

\bibitem{HG96} Henry Garrett, “SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions”, ResearchGate 2022 (doi: 10.13140/RG.2.2.20993.12640).

\bibitem{HG952}  Henry Garrett,“Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments”, Preprints 2022, 2022120324 (doi: 10.20944/preprints202212.0324.v1).

\bibitem{HG95} Henry Garrett, “Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs And SuperHyperGraphs Alongside Applications in Cancer’s Treatments”, ResearchGate 2022 (doi: 10.13140/RG.2.2.23123.04641).

\bibitem{HG942}  Henry Garrett, “SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses”, Preprints 2022, 2022110576 (doi: 10.20944/preprints202211.0576.v1).

\bibitem{HG94} Henry Garrett, “SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses”, ResearchGate 2022 (doi: 10.13140/RG.2.2.23324.56966).

\bibitem{HG37}  Henry Garrett, \textit{``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph''}, ResearchGate 2022 (doi: 10.13140/RG.2.2.29173.86244).

\bibitem{HG38} Henry Garrett, ``\textit{Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)}'', ResearchGate 2022 (doi: 10.13140/RG.2.2.25385.88160).

\bibitem{HG154b}
Henry Garrett, “Hamiltonian-Cycle-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7793875).

\bibitem{HG153b}
 Henry Garrett, “Hamiltonian-Cycle-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7792307).

\bibitem{HG152b} Henry Garrett, “Hamiltonian-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7790728).

\bibitem{HG151b} Henry Garrett, “Hamiltonian-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7787712).

\bibitem{HG150b} Henry Garrett, “Hamiltonian-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7783791).

\bibitem{HG149b} Henry Garrett, “Trace-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7780123).

\bibitem{HG148b} Henry Garrett, “Trace-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7773119).

\bibitem{HG147b} Henry Garrett, “SuperHyperDuality”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7637762).

\bibitem{HG146b} Henry Garrett, “Trace-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7766174).

\bibitem{HG145b}Henry Garrett, “Edge-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7762232).

\bibitem{HG144b} Henry Garrett, “Edge-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7758601).

\bibitem{HG143b} Henry Garrett, “Edge-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7754661).

\bibitem{HG142b} Henry Garrett, “Vertex-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7750995) .

\bibitem{HG141b} Henry Garrett, “Vertex-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7749875).

\bibitem{HG140b} Henry Garrett, “Vertex-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7747236).

\bibitem{HG139b} Henry Garrett, “Stable-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7742587).

\bibitem{HG138b} Henry Garrett, “Stable-Decompositions In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7738635).

\bibitem{HG137b} Henry Garrett, “Stable-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7734719).

\bibitem{HG136b} Henry Garrett, “Clique-Neighbors In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7730484).

\bibitem{HG135b} Henry Garrett, “Clique-Decompositions In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7730469).

\bibitem{HG134b} Henry Garrett, “Clique-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7722865).

\bibitem{HG133b} Henry Garrett, “Space In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7713563).

\bibitem{HG132b} Henry Garrett, “Space In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7709116).

\bibitem{HG131b} Henry Garrett, “Edge-Connectivity In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7706415).

\bibitem{HG130b} Henry Garrett, “Vertex-Connectivity In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7706063).

\bibitem{HG129b} Henry Garrett, “Tree-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7701906).

\bibitem{HG128b} Henry Garrett, “Chord In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7700205).

\bibitem{HG127b} Henry Garrett, “(i,j)-Domination In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7694876).

\bibitem{HG126b} Henry Garrett, “Edge-Domination In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7679410).

\bibitem{HG125b} Henry Garrett, “K-Domination In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7675982).

\bibitem{HG124b} Henry Garrett, “K-Number In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7672388).

\bibitem{HG123b} Henry Garrett, “Order In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7668648).

\bibitem{HG122b} Henry Garrett, “Coloring In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7662810).

\bibitem{HG121b} Henry Garrett, “Dimension In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7659162).

\bibitem{HG120b} Henry Garrett, “Cancer In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653233).

\bibitem{HG119b} Henry Garrett, “SuperHyperWheel”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653204).

\bibitem{HG118b} Henry Garrett, “SuperHyperMultipartite”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653142).

\bibitem{HG117b} Henry Garrett, “SuperHyperBipartite”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653117).

\bibitem{HG116b} Henry Garrett, “SuperHyperStar”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653089).

\bibitem{HG115b} Henry Garrett, “SuperHyperCycle”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651687).

\bibitem{HG114b} Henry Garrett, “SuperHyperPath”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651619).

\bibitem{HG113b} Henry Garrett, “SuperHyperDomination”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651439).

\bibitem{HG112b} Henry Garrett, “SuperHyperDominating”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7650729).

\bibitem{HG111b} Henry Garrett, “SuperHyperConnected”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7647868).

\bibitem{HG110b} Henry Garrett, “SuperHyperTotal”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7647017).

\bibitem{HG109b} Henry Garrett, “SuperHyperPerfect”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7644894).

\bibitem{HG108b} Henry Garrett, “SuperHyperJoin”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7641880).

\bibitem{HG107b} Henry Garrett, “Path SuperHyperColoring”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7632923).

\bibitem{HG106b} Henry Garrett, “SuperHyperDensity”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7623459).

\bibitem{HG105b} Henry Garrett, “Neutrosophic SuperHyperConnectivities”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606416).

\bibitem{HG104b} Henry Garrett, “Extreme SuperHyperConnectivities”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606416).

\bibitem{HG103b} Henry Garrett, “SuperHyperConnectivities”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606404).

\bibitem{HG102b} Henry Garrett, “Neutrosophic SuperHyperCycle”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).

\bibitem{HG101b} Henry Garrett, “Extreme SuperHyperCycle”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).

\bibitem{HG100b} Henry Garrett, “Extreme SuperHyperCycle”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).

\bibitem{HG99b} Henry Garrett, “SuperHyperCycle”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7579929).

\bibitem{HG98b} Henry Garrett, “Neutrosophic SuperHyperGirth”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563170).

\bibitem{HG97b} Henry Garrett, “Extreme SuperHyperGirth”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563164).
   
   \bibitem{HG96b} Henry Garrett, “SuperHyperGirth”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
   
   \bibitem{HG95b} Henry Garrett, “Extreme SuperHyperGirth”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
   
   \bibitem{HG94b} Henry Garrett, “Overlook On SuperHyperGirth”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
   
   \bibitem{HG93b} Henry Garrett, “Neutrosophic SuperHyperMatching”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7557063).
   
   \bibitem{HG92b} Henry Garrett, “Extreme SuperHyperMatching”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7557009).
   
   \bibitem{HG91b} Henry Garrett, “Overlook On SuperHyperMatching”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7539484).
   
   \bibitem{HG90b} Henry Garrett, “Neutrosophic Failed SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
   
   \bibitem{HG89b} Henry Garrett, “Extreme Failed SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
   
   \bibitem{HG88b} Henry Garrett, “Overlook On Failed SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
   
   \bibitem{HG87b} Henry Garrett, “Extreme SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7574952).
   
  \bibitem{HG86b} Henry Garrett, “Neutrosophic SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7574992).
 
   \bibitem{HG85b} Henry Garrett, “Extreme SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).
   
   \bibitem{HG84b} Henry Garrett, “Overlook On SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).
   
   \bibitem{HG83b} Henry Garrett, “Neutrosophic Failed SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
   
   \bibitem{HG82b} Henry Garrett, “Extreme Failed SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
   
   \bibitem{HG81b} Henry Garrett, “Overlook On Failed SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
   
   \bibitem{HG80b} Henry Garrett, “Neutrosophic SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
   
   \bibitem{HG79b} Henry Garrett, “Extreme SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
   
   \bibitem{HG78b} Henry Garrett, “Overlook On SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
   
 \bibitem{HG77b} Henry Garrett, “Neutrosophic Failed SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
 
  \bibitem{HG76b} Henry Garrett, “Extreme Failed SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
 
   \bibitem{HG75b} Henry Garrett, “Neutrosophic SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
   
  \bibitem{HG74b} Henry Garrett, “Extreme SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
 
   \bibitem{HG73b} Henry Garrett, “Overlook On SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
   
   \bibitem{HG72b} Henry Garrett, “Neutrosophic SuperHyperAlliances”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
   
   \bibitem{HG71b} Henry Garrett, “Extreme SuperHyperAlliances”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
   
 \bibitem{HG70b} Henry Garrett, “Overlook On SuperHyperAlliances”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
 
   \bibitem{HG69b} Henry Garrett, “SuperHyperMatching”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7539484).
   
   \bibitem{HG68b} Henry Garrett, “Failed SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
   
   \bibitem{HG67b} Henry Garrett, “SuperHyperClique”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).  
   
   \bibitem{HG66b} Henry Garrett, “Failed SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
   
 \bibitem{HG65b} Henry Garrett, “SuperHyperStable”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
 
   \bibitem{HG64b} Henry Garrett, “Failed SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
   
   \bibitem{HG63b} Henry Garrett, “SuperHyperForcing”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
   
 \bibitem{HG62b} Henry Garrett, “SuperHyperAlliances”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
 
\bibitem{HG61b} Henry Garrett, “SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7480110).

   \bibitem{HG60b} Henry Garrett, “Neut. SuperHyperEdges”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7378758).
   
 \bibitem{HG32b} Henry Garrett, “Beyond Neutrosophic Graphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.6320305).
   
 \bibitem{HG44b} Henry Garrett, “Neutrosophic Duality”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.6677173).

\end{thebibliography}
\end{document}