New Ideas On Super connections By Hyper disconnections Of SuperHyperJoin In Cancer's Recognition with (Neutrosophic) SuperHyperGraph
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- 1. Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA
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DOI: 10.13140/RG.2.2.33275.80161
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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).
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\fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
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New Ideas On Super connections By Hyper disconnections Of SuperHyperJoin In Cancer's Recognition with (Neutrosophic) SuperHyperGraph
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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 SuperHyperJoin). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is an ordered 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-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 $V_a\not\in E_i,E_j;$ 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 $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ 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;$ 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 $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyperJoin if it's either of Neutrosophic e-SuperHyperJoin, Neutrosophic re-SuperHyperJoin, Neutrosophic v-SuperHyperJoin, and Neutrosophic rv-SuperHyperJoin. ((Neutrosophic) SuperHyperJoin).
Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is an ordered pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called an 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 consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; a 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 consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; an 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 consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient; a 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 consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an 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 consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; a 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 consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; an 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 consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient; a 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 consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperJoin; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyperJoin and Extreme SuperHyperJoin. 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 ``Extreme 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 $\delta-$SuperHyperJoin is a maximal of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (Extreme) 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; an Extreme $\delta-$SuperHyperJoin is a maximal Extreme of SuperHyperVertices with maximum Extreme cardinality such that either of the following expressions hold for the Extreme cardinalities of SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{Extreme} > |S\cap (V\setminus N(s))|_{Extreme}+\delta;$ and
$ |S\cap N(s)|_{Extreme} < |S\cap (V\setminus N(s))|_{Extreme}+\delta.$
The first Expression, holds if $S$ is an Extreme $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an Extreme $\delta-$SuperHyperDefensive
It's useful to define a ``Extreme'' version of a SuperHyperJoin . Since there's more ways to get type-results to make a SuperHyperJoin more understandable. For the sake of having Extreme SuperHyperJoin, there's a need to ``redefine'' the notion of a ``SuperHyperJoin ''. 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 SuperHyperJoin . It's redefined an Extreme SuperHyperJoin if the mentioned Table holds, concerning, ``The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme 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 SuperHyperJoin . 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 SuperHyperJoin until the SuperHyperJoin, then it's officially called a ``SuperHyperJoin'' but otherwise, it isn't a SuperHyperJoin . There are some instances about the clarifications for the main definition titled a ``SuperHyperJoin ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyperJoin . For the sake of having an Extreme SuperHyperJoin, there's a need to ``redefine'' the notion of a ``Extreme SuperHyperJoin'' and a ``Extreme SuperHyperJoin ''. 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 an Extreme SuperHyperGraph. It's redefined ``Extreme SuperHyperGraph'' if the intended Table holds. And a SuperHyperJoin are redefined to a ``Extreme SuperHyperJoin'' if the intended Table holds. It's useful to define ``Extreme'' version of SuperHyperClasses. Since there's more ways to get Extreme type-results to make an Extreme SuperHyperJoin more understandable. Assume an Extreme SuperHyperGraph. There are some Extreme SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are
``Extreme SuperHyperPath'', ``Extreme SuperHyperJoin'', ``Extreme SuperHyperStar'', ``Extreme SuperHyperBipartite'', ``Extreme SuperHyperMultiPartite'', and ``Extreme SuperHyperWheel'' if the intended Table holds. A SuperHyperGraph has a ``Extreme SuperHyperJoin'' where it's the strongest [the maximum Extreme value from all the SuperHyperJoin amid the maximum value amid all SuperHyperVertices from a SuperHyperJoin .] SuperHyperJoin . A graph is a SuperHyperUniform if it's a SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume an Extreme 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 SuperHyperJoin 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 ``Extreme 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 ``Extreme''. 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 Extreme 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 an Extreme SuperHyperPath(-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the longest SuperHyperJoin or the strongest SuperHyperJoin in those Extreme SuperHyperModels. For the longest SuperHyperJoin, called SuperHyperJoin, and the strongest SuperHyperJoin, called Extreme SuperHyperJoin, 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 SuperHyperJoin. There isn't any formation of any SuperHyperJoin but literarily, it's the deformation of any SuperHyperJoin. It, literarily, deforms and it doesn't form. A basic familiarity with Extreme SuperHyperJoin theory, SuperHyperGraphs, and Extreme SuperHyperGraphs theory are proposed.
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\textbf{Keywords:} Extreme SuperHyperGraph, SuperHyperJoin, Cancer's Extreme Recognition
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\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45
\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 date back on January 22, 2023.
\\
First article is titled ``properties of SuperHyperGraph and neutrosophic SuperHyperGraph'' in \textbf{Ref.} \cite{HG1} by Henry Garrett (2022). It's first step toward the research on neutrosophic SuperHyperGraphs. This research article is published on the journal ``Neutrosophic Sets and Systems'' in issue 49 and the pages 531-561. In this research article, different types of notions like dominating, resolving, coloring, Eulerian(Hamiltonian) neutrosophic path, n-Eulerian(Hamiltonian) neutrosophic path, zero forcing number, zero forcing neutrosophic- number, independent number, independent neutrosophic-number, clique number, clique neutrosophic-number, matching number, matching neutrosophic-number, girth, neutrosophic girth, 1-zero-forcing number, 1-zero- forcing neutrosophic-number, failed 1-zero-forcing number, failed 1-zero-forcing neutrosophic-number, global- offensive alliance, t-offensive alliance, t-defensive alliance, t-powerful alliance, and global-powerful alliance are defined in SuperHyperGraph and neutrosophic SuperHyperGraph. Some Classes of SuperHyperGraph and Neutrosophic SuperHyperGraph are cases of research. Some results are applied in family of SuperHyperGraph and neutrosophic SuperHyperGraph. Thus this research article has concentrated on the vast notions and introducing the majority of notions.
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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{HG2} 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 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.
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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{HG3} 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 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.
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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), there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph.
\\
Some scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG39} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 3230 readers in Scribd. It's titled ``Beyond Neutrosophic Graphs'' and published by Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United State. 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{HG40} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 4117 readers in Scribd. It's titled ``Neutrosophic Duality'' and published by Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. 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 Extreme SuperHyperJoin theory, Neutrosophic SuperHyperJoin 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}. Two popular scientific research books in Scribd in the terms of high readers, 3230 and 4117 respectively, on neutrosophic science is on \cite{HG39,HG40}.
\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 ``Extreme 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 ``Extreme 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 Extreme 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 `` SuperHyperJoin'' 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 Extreme 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 an Extreme SuperHyperPath (-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the optimal SuperHyperJoin or the Extreme SuperHyperJoin in those Extreme SuperHyperModels. Some general results are introduced. Beyond that in SuperHyperStar, all possible Extreme 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 SuperHyperJoin. There isn't any formation of any SuperHyperJoin but literarily, it's the deformation of any SuperHyperJoin. 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 SuperHyperJoin'' 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 SuperHyperJoin'' 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 `` SuperHyperJoin'' and ``Extreme SuperHyperJoin'' on ``SuperHyperGraph'' and ``Extreme 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 Extreme 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, SuperHyperJoin and Extreme SuperHyperJoin, are figured out in sections `` SuperHyperJoin'' and ``Extreme SuperHyperJoin''. In the sense of tackling on getting results and in order to make sense about continuing the research, the ideas of SuperHyperUniform and Extreme 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 Extreme 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 Extreme SuperHyperGraph, in the sections ``Results on SuperHyperClasses'' and ``Results on Extreme 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'',
`` SuperHyperJoin'', ``Extreme SuperHyperJoin'', ``Results on SuperHyperClasses'' and ``Results on Extreme 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, `` SuperHyperJoin''. 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{Extreme 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 space 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 space 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 an ordered 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 an ordered 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 an ordered 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 an ordered 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 an ordered 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 an ordered 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 an ordered 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 an ordered 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 SuperHyperJoin).\\
Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is an ordered 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 an ordered 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 consecutive 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 consecutive 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 consecutive 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 consecutive 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 consecutive 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 consecutive 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 consecutive 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 consecutive 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{definition}((Extreme/Neutrosophic)$\delta-$SuperHyperJoin).\\
Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is an ordered pair $S=(V,E).$ Then
\begin{itemize}
\item[$(i)$] an \textbf{$\delta-$SuperHyperJoin} is a Neutrosophic kind of Neutrosophic SuperHyperJoin 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 $\delta-$SuperHyperJoin} is a Neutrosophic kind of Neutrosophic SuperHyperJoin 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 SuperHyperJoin, 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 an ordered 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 an ordered pair $S=(V,E).$ There are some \textbf{Neutrosophic SuperHyperClasses} if the Table \eqref{136TBL4} holds. Thus Neutrosophic SuperHyperPath , SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are
\textbf{Neutrosophic SuperHyperPath }, \textbf{Neutrosophic SuperHyperJoin}, \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 SuperHyperJoin. Since there's more ways to get type-results to make a Neutrosophic SuperHyperJoin more Neutrosophicly understandable.
\\
For the sake of having a Neutrosophic SuperHyperJoin, there's a need to ``\textbf{redefine}'' the Neutrosophic notion of ``Neutrosophic SuperHyperJoin''. 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 SuperHyperJoin. It's redefined a \textbf{Neutrosophic SuperHyperJoin} 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{
Extreme SuperHyperJoin But As The Extensions Excerpt From Dense And Super Forms}
\begin{example}\label{136EXM1}
Assume an Extreme SuperHyperGraph (NSHG) $S$ is an ordered pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward. $E_1$ and $E_3$ are some empty Extreme
SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, namely, $E_4.$ The Extreme SuperHyperVertex, $V_3$ is Extreme isolated means that there's no Extreme SuperHyperEdge has it as an Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Extreme SuperHyperJoin.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG1.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG1}
\end{figure}
\item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, namely, $E_4.$ The Extreme SuperHyperVertex, $V_3$ is Extreme isolated means that there's no Extreme SuperHyperEdge has it as an Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Extreme SuperHyperJoin.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG2.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG2}
\end{figure}
\item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG3.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG3}
\end{figure}
\item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG4.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG4}
\end{figure}
\item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG5.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG5}
\end{figure}
\item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}6z^8.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^7}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG6.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG6}
\end{figure}
\item On the Figure \eqref{136NSHG7}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{15},E_{16},E_{17}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^3.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG7.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG7}
\end{figure}
\item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG8.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG8}
\end{figure}
\item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^5.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG9.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG9}
\end{figure}
\item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG10.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG10}
\end{figure}
\item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1,E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_6,V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
3\times3z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG11.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG11}
\end{figure}
\item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG12.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG12}
\end{figure}
\item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_3,E_9\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
3\times3 z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG13.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG13}
\end{figure}
\item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG14.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG14}
\end{figure}
\item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG15.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG15}
\end{figure}
\item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG16.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG16}
\end{figure}
\item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG17.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG17}
\end{figure}
\item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG18}
\end{figure}
\item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0^3}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^4.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{2i+1}_{i=0^5}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG19}
\end{figure}
\item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_6\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=10z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{136NSHG20}
\end{figure}
\item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{95NHG1.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{95NHG1}
\end{figure}
\item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{95NHG2.png}
\caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM1} }
\label{95NHG2}
\end{figure}
\end{itemize}
\end{example}
\section{The Extreme Departures on The Theoretical Results Toward Theoretical Motivations}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme 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{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected Extreme 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 Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{an Extreme SuperHyperPath Associated to the Notions of Extreme SuperHyperJoin in the Example \eqref{136EXM18a}}
\label{136NSHG18a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme 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{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected Extreme 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 Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{an Extreme SuperHyperCycle Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM19a}}
\label{136NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperJoin}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme 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 Extreme 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 Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperStar $ESHS:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG20a}, is the Extreme SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{an Extreme SuperHyperStar Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme Example \eqref{136EXM20a}}
\label{136NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme 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{Extreme 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{Extreme 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{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme 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{Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme 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 Extreme 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 Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-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 Extreme Figure \eqref{136NSHG21a}, the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ is Extreme highlighted and Extreme featured. The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG21a}, is the Extreme SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{Extreme SuperHyperBipartite Extreme Associated to the Extreme Notions of Extreme SuperHyperJoin in the Example \eqref{136EXM21a}}
\label{136NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme 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{Extreme 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{Extreme 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{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme 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{Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme 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 Extreme 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 Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-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 Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Extreme featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG22a}, is the Extreme SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{an Extreme SuperHyperMultipartite Associated to the Notions of Extreme SuperHyperJoin in the Example \eqref{136EXM22a}}
\label{136NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme 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{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
is a longest SuperHyperJoin taken from a connected Extreme 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 Extreme Figure \eqref{136NSHG23a}, the connected Extreme SuperHyperWheel $NSHW:(V,E),$ is Extreme highlighted and featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperWheel $ESHW:(V,E),$
in the Extreme SuperHyperModel \eqref{136NSHG23a}, is the Extreme SuperHyperJoin.
\begin{figure}
\includegraphics[width=100mm]{136NSHG23.png}
\caption{an Extreme SuperHyperWheel Extreme Associated to the Extreme Notions of Extreme SuperHyperJoin in the Extreme 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 SuperHyperJoin, Extreme SuperHyperJoin, and the Extreme SuperHyperJoin, some general results are introduced.
\begin{remark}
Let remind that the Extreme SuperHyperJoin is ``redefined'' on the positions of the alphabets.
\end{remark}
\begin{corollary}
Assume Extreme SuperHyperJoin. Then
\begin{eqnarray*}
&& Extreme ~SuperHyperJoin=\\&&\{the SuperHyperJoin of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive \\&&SuperHyperJoin
\\&&
|_{Extreme cardinality amid those SuperHyperJoin.}\}
\end{eqnarray*}
plus one Extreme 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 an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then the notion of Extreme SuperHyperJoin and SuperHyperJoin coincide.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is an Extreme SuperHyperJoin if and only if it's a SuperHyperJoin.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a strongest SuperHyperJoin if and only if it's a longest SuperHyperJoin.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then its Extreme SuperHyperJoin is its SuperHyperJoin and reversely.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperPath(-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel) on the same identical letter of the alphabet. Then its Extreme SuperHyperJoin is its SuperHyperJoin and reversely.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperGraph. Then its Extreme SuperHyperJoin isn't well-defined if and only if its SuperHyperJoin isn't well-defined.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of an Extreme SuperHyperGraph. Then its Extreme SuperHyperJoin isn't well-defined if and only if its SuperHyperJoin isn't well-defined.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperPath(-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Extreme SuperHyperJoin isn't well-defined if and only if its SuperHyperJoin isn't well-defined.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperGraph. Then its Extreme SuperHyperJoin is well-defined if and only if its SuperHyperJoin is well-defined.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of an Extreme SuperHyperGraph. Then its Extreme SuperHyperJoin is well-defined if and only if its SuperHyperJoin is well-defined.
\end{corollary}
\begin{corollary}
Assume an Extreme SuperHyperPath(-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Extreme SuperHyperJoin is well-defined if and only if its SuperHyperJoin is well-defined.
\end{corollary}
%
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then $V$ is
\begin{itemize}
\item[$(i):$] the dual SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] the strong dual SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] the connected dual SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] the $\delta$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] the strong $\delta$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] the connected $\delta$-dual SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $NTG:(V,E,\sigma,\mu)$ be an Extreme SuperHyperGraph. Then $\emptyset$ is
\begin{itemize}
\item[$(i):$] the SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] the strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] the connected defensive SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then an independent SuperHyperSet is
\begin{itemize}
\item[$(i):$] the SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] the strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] the connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperJoin/SuperHyperPath. Then $V$ is a maximal
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperJoin;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is a SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal
\begin{itemize}
\item[$(i):$] dual SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperJoin;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperJoin/SuperHyperPath. Then the number of
\begin{itemize}
\item[$(i):$] the SuperHyperJoin;
\item[$(ii):$] the SuperHyperJoin;
\item[$(iii):$] the connected SuperHyperJoin;
\item[$(iv):$] the $\mathcal{O}(ESHG)$-SuperHyperJoin;
\item[$(v):$] the strong $\mathcal{O}(ESHG)$-SuperHyperJoin;
\item[$(vi):$] the connected $\mathcal{O}(ESHG)$-SuperHyperJoin.
\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 an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of
\begin{itemize}
\item[$(i):$] the dual SuperHyperJoin;
\item[$(ii):$] the dual SuperHyperJoin;
\item[$(iii):$] the dual connected SuperHyperJoin;
\item[$(iv):$] the dual $\mathcal{O}(ESHG)$-SuperHyperJoin;
\item[$(v):$] the strong dual $\mathcal{O}(ESHG)$-SuperHyperJoin;
\item[$(vi):$] the connected dual $\mathcal{O}(ESHG)$-SuperHyperJoin.
\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 an Extreme 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 SuperHyperJoin;
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme 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 SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\delta$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\delta$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize}
\item[$(i):$] dual SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperJoin.
\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 an Extreme SuperHyperGraph. The number of connected component is $|V-S|$ if there's a SuperHyperSet which is a dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] SuperHyperJoin;
\item[$(v):$] strong 1-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected 1-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and the Extreme number is at most $\mathcal{O}_n(ESHG).$
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the Extreme 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 SuperHyperJoin;
\item[$(ii):$] strong
SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is $\emptyset.$ The number is $0$ and the Extreme number is $0,$ for an independent SuperHyperSet in the setting of dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $0$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $0$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $0$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperComplete. Then there's no independent SuperHyperSet.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperJoin/SuperHyperPath/SuperHyperWheel. The number is $\mathcal{O}(ESHG:(V,E))$ and the Extreme number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the Extreme 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 SuperHyperJoin;
\item[$(ii):$] strong SuperHyperDefensive SuperHyperJoin;
\item[$(iii):$] connected SuperHyperDefensive SuperHyperJoin;
\item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin;
\item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin;
\item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ Extreme 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 Extreme SuperHyperGraphs.
\end{proposition}
%
\begin{proposition}
Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperJoin, 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 Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperJoin, then
\begin{itemize}
\item[$(i)$]
$S$ is SuperHyperDominating 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 Extreme 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 Extreme 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 SuperHyperJoin;
\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 SuperHyperJoin.
\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 SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an even SuperHyperJoin. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is a dual SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an odd SuperHyperJoin. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\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 SuperHyperJoin;
\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 SuperHyperJoin.
\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 SuperHyperJoin;
\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 SuperHyperJoin.
\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 SuperHyperJoin;
\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 SuperHyperJoin.
\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 SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of Extreme SuperHyperStars with common Extreme SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is a dual SuperHyperDefensive SuperHyperJoin 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 SuperHyperJoin 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 Extreme 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 SuperHyperJoin 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 SuperHyperJoin 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 Extreme SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual SuperHyperDefensive SuperHyperJoin 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 SuperHyperJoin for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then following statements hold;
\begin{itemize}
\item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperJoin, then $S$ is an s-SuperHyperDefensive SuperHyperJoin;
\item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperJoin, then $S$ is a dual s-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then following statements hold;
\begin{itemize}
\item[$(i)$] if $s\geq t+2$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperJoin, then $S$ is an s-SuperHyperPowerful SuperHyperJoin;
\item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperJoin, then $S$ is a dual s-SuperHyperPowerful SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a[an] [r-]SuperHyperUniform-strong-Extreme 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 SuperHyperJoin;
\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 SuperHyperJoin;
\item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an r-SuperHyperDefensive SuperHyperJoin;
\item[$(iv)$] if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual r-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-Extreme 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 SuperHyperJoin;
\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 SuperHyperJoin;
\item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an r-SuperHyperDefensive SuperHyperJoin;
\item[$(iv)$] $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual r-SuperHyperDefensive SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-Extreme 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 SuperHyperJoin;
\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 SuperHyperJoin;
\item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-Extreme 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 SuperHyperJoin;
\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 SuperHyperJoin;
\item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperJoin. 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 SuperHyperJoin;
\item[$(ii)$] $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperJoin;
\item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperJoin. 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 SuperHyperJoin;
\item[$(ii)$] if $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperJoin;
\item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyperJoin;
\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 SuperHyperJoin.
\end{itemize}
\end{proposition}
\section{Extreme Applications in Cancer's Extreme Recognition}
The cancer is the Extreme disease but the Extreme model is going to figure out what's going on this Extreme phenomenon. The special Extreme case of this Extreme 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 Extreme recognition of the cancer could help to find some Extreme treatments for this Extreme disease.
\\
In the following, some Extreme steps are Extreme devised on this disease.
\begin{description}
\item[Step 1. (Extreme Definition)] The Extreme recognition of the cancer in the long-term Extreme function.
\item[Step 2. (Extreme Issue)] The specific region has been assigned by the Extreme model [it's called Extreme SuperHyperGraph] and the long Extreme 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 Extreme SuperHyperGraph] to have convenient perception on what's happened and what's done.
\item[Step 3. (Extreme Model)]
There are some specific Extreme models, which are well-known and they've got the names, and some general Extreme models. The moves and the Extreme traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by an Extreme SuperHyperPath(-/SuperHyperJoin, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the Extreme SuperHyperJoin or the Extreme SuperHyperJoin in those Extreme Extreme SuperHyperModels.
\section{Case 1: The Initial Extreme Steps Toward Extreme SuperHyperBipartite as Extreme SuperHyperModel}
\item[Step 4. (Extreme Solution)]
In the Extreme Figure \eqref{136NSHGaa21aa}, the Extreme SuperHyperBipartite is Extreme highlighted and Extreme featured.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{an Extreme SuperHyperBipartite Associated to the Notions of Extreme SuperHyperJoin}
\label{136NSHGaa21aa}
\end{figure}
\\ By using the Extreme Figure \eqref{136NSHGaa21aa} and the Table \eqref{136TBLaa21aa}, the Extreme SuperHyperBipartite is obtained.
\\
The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHGaa21aa},
is the Extreme SuperHyperJoin.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme 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 Extreme Steps Toward Extreme SuperHyperMultipartite as Extreme SuperHyperModel}
\item[Step 4. (Extreme Solution)]
In the Extreme Figure \eqref{136NSHGaa22aa}, the Extreme SuperHyperMultipartite is Extreme highlighted and Extreme featured.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{an Extreme SuperHyperMultipartite Associated to the Notions of Extreme SuperHyperJoin}
\label{136NSHGaa22aa}
\end{figure}
\\
By using the Extreme Figure \eqref{136NSHGaa22aa} and the Table \eqref{136TBLaa22aa}, the Extreme SuperHyperMultipartite is obtained.
\\
The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHGaa22aa}, is the Extreme SuperHyperJoin.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme 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 SuperHyperJoin and the Extreme SuperHyperJoin 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 SuperHyperJoin and the Extreme SuperHyperJoin?
\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 SuperHyperJoin and the Extreme SuperHyperJoin?
\end{question}
\begin{problem}
The SuperHyperJoin and the Extreme SuperHyperJoin do a SuperHyperModel for the Cancer's recognitions and they're based on SuperHyperJoin, 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 Extreme SuperHyperGraphs more understandable. In this endeavor, two SuperHyperNotions are defined on the SuperHyperJoin. For that sake in the second definition, the main definition of the Extreme SuperHyperGraph is redefined on the position of the alphabets. Based on the new definition for the Extreme SuperHyperGraph, the new SuperHyperNotion, Extreme SuperHyperJoin, finds the convenient background to implement some results based on that. Some SuperHyperClasses and some Extreme 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, SuperHyperJoin, the new SuperHyperClasses and SuperHyperClasses, are introduced. Some general results are gathered in the section on the SuperHyperJoin and the Extreme SuperHyperJoin. 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 Extreme 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 `` SuperHyperJoin'' 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 Extreme SuperHyperGraph} &\textcolor{black}{1. }\textcolor{blue}{General Results} \\ &
\\
\textcolor{black}{2. }\textcolor{red}{ SuperHyperJoin}& \\ &
\\
\textcolor{black}{3. } \textcolor{red}{Extreme SuperHyperJoin} &\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.
\begin{thebibliography}{595}
\bibitem{HG1} Henry Garrett, ``\textit{Properties of SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Neutrosophic Sets and Systems 49 (2022) 531-561 (doi: 10.5281/zenodo.6456413). (http://fs.unm.edu/NSS/NeutrosophicSuperHyperGraph34.pdf). (https://digitalrepository.unm.edu/nss\_journal/vol49/iss1/34).
\bibitem{HG2} 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{HG3} Henry Garrett, ``\textit{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.
\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{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{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{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{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{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{HG39} Henry Garrett, (2022). ``\textit{Beyond Neutrosophic Graphs}'', Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United States. ISBN: 979-1-59973-725-6 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf).
\bibitem{HG40} Henry Garrett, (2022). ``\textit{Neutrosophic Duality}'', Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 (http://fs.unm.edu/NeutrosophicDuality.pdf).
\end{thebibliography}
\end{document}
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