New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Cut As Hyper Eulogy-Path-Cut On Super EULA-Path-Cut

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\fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
\fancyfoot[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
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\textbf\newline{
New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Cut As Hyper Eulogy-Path-Cut On Super EULA-Path-Cut
}
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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, some extreme notions and Neutrosophic notions are defined on the family of SuperHyperGraphs and Neutrosophic SuperHyperGraphs. Some well-known classes are used in this scientific research. A basic familiarity with Neutrosophic  SuperHyper Eulerian-Path-Cut theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.
\\ \vspace{4mm}
\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyperEulerian-Path-Cut, Cancer's Neutrosophic Recognition
   \\
\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45

\section{Neutrosophic Preliminaries Of This Scientific Research On the Redeemed Ways}
In this section, the basic material in this scientific research, is referred to [Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2), [Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1), [Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.5,p.2), [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [t-norm](\textbf{Ref.}\cite{HG38}, Definition 2.7, p.3), and [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [Neutrosophic Strength of the Neutrosophic SuperHyperPaths] (\textbf{Ref.}\cite{HG38},Definition 5.3,p.7), and [Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)] (\textbf{Ref.}\cite{HG38},Definition 5.4,p.7). Also, the new ideas and their clarifications are addressed to \textbf{Ref.}\cite{HG38}.

\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperEulerian-Path-Cut).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic e-SuperHyperEulerian-Path-Cut criteria} holds
  \begin{eqnarray*}
&&
\forall E_a\in P: P~\text{is}
\\&&
\text{a SuperHyperPath and it has}
\\&&
 \text{the all number of SuperHyperEdges};
\end{eqnarray*}

\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic re-SuperHyperEulerian-Path-Cut criteria} holds
  \begin{eqnarray*}
&&
\forall E_a\in P: P~\text{is}
\\&&
\text{a SuperHyperPath and it has}
\\&&
 \text{the all number of SuperHyperEdges};
\end{eqnarray*}

and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut criteria} holds
  \begin{eqnarray*}
&&
\forall  V_a\in P: P~\text{is}
\\&&
\text{a SuperHyperPath and it has}
\\&&
 \text{the all number of SuperHyperEdges};
\end{eqnarray*}

\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut criteria} holds
  \begin{eqnarray*}
&&
\forall  V_a\in P: P~\text{is}
\\&&
\text{a SuperHyperPath and it has}
\\&&
 \text{the all number of SuperHyperEdges};
\end{eqnarray*}
and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperEulerian-Path-Cut).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme V-SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut;
 \item[$(vi)$]
a \textbf{Neutrosophic V-SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut;
 \item[$(vii)$]
an \textbf{Extreme V-SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is  the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}
\section{
Neutrosophic SuperHyperEulerian-Path-Cut But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Neutrosophic event).\\
  Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Any Neutrosophic k-subset of $A$ of $V$ is called \textbf{Neutrosophic k-event} and if $k=2,$ then Neutrosophic subset of $A$ of $V$ is called \textbf{Neutrosophic event}. The following expression is called \textbf{Neutrosophic probability} of $A.$
\begin{eqnarray}
 E(A)=\sum_{a\in A}E(a).
\end{eqnarray}
\end{definition}
\begin{definition}(Neutrosophic Independent).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. $s$ Neutrosophic k-events $A_i,~i\in I$ is called \textbf{Neutrosophic s-independent} if the following expression is called \textbf{Neutrosophic s-independent criteria}
\begin{eqnarray*}
 E(\cap_{i\in I}A_i)=\prod_{i\in I}P(A_i).
\end{eqnarray*}
And if $s=2,$ then Neutrosophic k-events of $A$ and $B$ is called \textbf{Neutrosophic independent}. The following expression is called \textbf{Neutrosophic independent criteria}
\begin{eqnarray}
 E(A\cap B)=P(A)P(B).
\end{eqnarray}
\end{definition}
\begin{definition}(Neutrosophic Variable).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Any k-function Eulerian-Path-Cut like $E$ is called \textbf{Neutrosophic k-Variable}. If $k=2$, then any 2-function Eulerian-Path-Cut like $E$ is called \textbf{Neutrosophic Variable}.
\end{definition}
The notion of independent on Neutrosophic Variable is likewise.
\begin{definition}(Neutrosophic Expectation).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. A Neutrosophic k-Variable $E$ has a number is called \textbf{Neutrosophic Expectation} if the following expression is called \textbf{Neutrosophic Expectation criteria}
\begin{eqnarray*}
 Ex(E)=\sum_{\alpha \in V}E(\alpha)P(\alpha).
\end{eqnarray*}
\end{definition}
\begin{definition}(Neutrosophic Crossing).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. A Neutrosophic number is called \textbf{Neutrosophic Crossing} if the following expression is called \textbf{Neutrosophic Crossing criteria}
\begin{eqnarray*}
 Cr(S)=\min\{\text{Number of Crossing in a Plane Embedding of } S\}.
\end{eqnarray*}
\end{definition}
\begin{lemma}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $m$ and $n$ propose special Eulerian-Path-Cut. Then with $m\geq 4n,$

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

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

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

\begin{corollary}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $G \in \mathcal{G}_{n,1/2}$ and let $f$ and $k^{*}$ be as defined in previous Theorem. Then either:
   \begin{itemize}
 \item[$(i).$] $f(k^{*}) << 1,$ in which case almost surely $\alpha(G)$ is equal to either  $k^{*}-2$ or  $k^{*}-1$, or
 \item[$(ii).$] $f(k^{*}-1) >> 1,$ in which case almost surely $\alpha(G)$  is equal to either $k^{*}-1$ or $k^{*}.$
\end{itemize}
\end{corollary}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. The latter is straightforward.
\end{proof}
\begin{definition}(Neutrosophic Threshold).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $P$ be a  monotone property of SuperHyperGraphs (one which is preserved when SuperHyperEdges are added). Then a \textbf{Neutrosophic Threshold} for $P$ is a function $f(n)$ such that:
 \begin{itemize}
 \item[$(i).$]  if $p << f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely does not have $P,$
 \item[$(ii).$]  if $p >> f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely has $P.$
\end{itemize}
\end{definition}
\begin{definition}(Neutrosophic Balanced).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $F$ be a fixed Neutrosophic SuperHyperGraph. Then there is a threshold function for the property of containing a copy of $F$ as a Neutrosophic SubSuperHyperGraph is called \textbf{Neutrosophic Balanced}.
\end{definition}
 \begin{theorem}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $F$ be a nonempty balanced Neutrosophic SubSuperHyperGraph with $k$ SuperHyperVertices and $l$ SuperHyperEdges. Then $n^{-k/l}$ is a threshold function for the property of containing $F$ as a Neutrosophic SubSuperHyperGraph.
\end{theorem}
 \begin{proof}
   Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Neutrosophic Figures in every Neutrosophic items.
 \begin{itemize}

 \item On the Figure \eqref{136NSHG4}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{E_4,E_5,E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =2z^5+2z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{V_i\in V_{NSHG}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =\sum z^{|V_i\in V_{NSHG}|}.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG4.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG4}
 \end{figure}
  \item On the Figure \eqref{136NSHG9}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG9.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG9}
 \end{figure}
 \item On the Figure \eqref{136NSHG10}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG10.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG10}
 \end{figure}
    \item On the Figure \eqref{136NSHG16}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{E_1,E_2,E_3,E_4,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =\sum z^{|E_i\in E_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{V_i\in V_{NSHG}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =\sum z^{|V_i\in V_{NSHG}|}.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG16.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG16}
 \end{figure}
  \item On the Figure \eqref{136NSHG20}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG20.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{136NSHG20}
 \end{figure}
 \item On the Figure \eqref{95NHG1}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG1.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{95NHG1}
 \end{figure}
 \item On the Figure \eqref{95NHG2}, the Neutrosophic SuperHyperNotion, namely, Neutrosophic SuperHyperEulerian-Path-Cut, is up. The Neutrosophic Algorithm is Neutrosophicly straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{E_2,E_3,E_4,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =\sum z^{|E_i\in E_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{V_i\in V_{NSHG}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =\sum z^{|V_i\in V_{NSHG}|}.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG2.png}  
   \caption{The Neutrosophic SuperHyperGraphs Associated to the Neutrosophic Notions of Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM1} }
 \label{95NHG2}
 \end{figure}
\end{itemize}
\end{example}
\section{The Neutrosophic Departures on The Theoretical Results Toward Theoretical Motivations}
The previous Neutrosophic approach apply on the upcoming Neutrosophic results on Neutrosophic SuperHyperClasses.
\begin{proposition}
Assume a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{E_i\in E_{NSHG}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =\sum z^{|E_i\in E_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{V_i\in V_{NSHG}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =\sum z^{|V_i\in V_{NSHG}|}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
be a longest path taken from a connected Neutrosophic SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperEulerian-Path-Cut. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Neutrosophic SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Neutrosophic SuperHyperSet,  in the Neutrosophic SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{a Neutrosophic SuperHyperPath Associated to the Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Example \eqref{136EXM18a}}
\label{136NSHG18a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  be a longest path taken from a connected Neutrosophic 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 SuperHyperEulerian-Path-Cut. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, in the Neutrosophic SuperHyperModel \eqref{136NSHG19a},
is the Neutrosophic  SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{a Neutrosophic SuperHyperCycle Associated to the Neutrosophic Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM19a}}
\label{136NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&CENTER,E_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,CENTER
 \end{eqnarray*}
be a longest path taken a connected Neutrosophic SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperEulerian-Path-Cut. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the  Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperStar $ESHS:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG20a}, is the  Neutrosophic SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{a Neutrosophic SuperHyperStar Associated to the Neutrosophic Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM20a}}
\label{136NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
  is a longest path taken from a connected   Neutrosophic SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperEulerian-Path-Cut. The latter is straightforward. Then there's no at least one SuperHyperEulerian-Path-Cut. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperEulerian-Path-Cut could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperEulerian-Path-Cut taken from a connected Neutrosophic SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Neutrosophic Figure \eqref{136NSHG21a}, the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ is Neutrosophic highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Neutrosophic Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperBipartite $ESHB:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG21a}, is the Neutrosophic  SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{Neutrosophic SuperHyperBipartite Neutrosophic Associated to the Neutrosophic Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Example \eqref{136EXM21a}}
\label{136NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
 is a longest SuperHyperEulerian-Path-Cut taken from a connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperEulerian-Path-Cut. The latter is straightforward.  Then there's no at least one SuperHyperEulerian-Path-Cut. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperEulerian-Path-Cut could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Neutrosophic-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Neutrosophic SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Neutrosophic featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous Neutrosophic result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperMultipartite $ESHM:(V,E),$ in the Neutrosophic SuperHyperModel \eqref{136NSHG22a}, is the  Neutrosophic SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{a Neutrosophic  SuperHyperMultipartite Associated to the Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Example \eqref{136EXM22a}}
\label{136NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut}}
           \\&&
          =\{\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Neutrosophic Eulerian-Path-Cut SuperHyperPolynomial}}
 \\&&
 =0z^0.
 \\&&
   \mathcal{C}(NSHG)_{\text{Neutrosophic V-Eulerian-Path-Cut}}
    \\&&
 =\{\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Neutrosophic V-Eulerian-Path-Cut SuperHyperPolynomial}}}
   \\&&
 =0z^0.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E^{*}_1,
 \\&&CENTER,E^{*}_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E^{*}_1,V^{EXTERNAL}_1,
 \\&&E^{*}_2,CENTER
 \end{eqnarray*}
  is a longest SuperHyperEulerian-Path-Cut taken from a connected Neutrosophic SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperEulerian-Path-Cut. The latter is straightforward. Then there's at least one SuperHyperEulerian-Path-Cut. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperEulerian-Path-Cut could be applied. The unique embedded SuperHyperEulerian-Path-Cut proposes some longest SuperHyperEulerian-Path-Cut excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Neutrosophic Figure \eqref{136NSHG23a}, the connected Neutrosophic SuperHyperWheel $NSHW:(V,E),$ is Neutrosophic highlighted and featured. The obtained Neutrosophic SuperHyperSet, by the Algorithm in previous result, of the Neutrosophic SuperHyperVertices of the connected Neutrosophic SuperHyperWheel $ESHW:(V,E),$
in the Neutrosophic SuperHyperModel \eqref{136NSHG23a}, is the  Neutrosophic SuperHyperEulerian-Path-Cut.
\begin{figure}
\includegraphics[width=100mm]{136NSHG23.png}
\caption{a Neutrosophic SuperHyperWheel Neutrosophic Associated to the Neutrosophic Notions of  Neutrosophic SuperHyperEulerian-Path-Cut in the Neutrosophic Example \eqref{136EXM23a}}
\label{136NSHG23a}
\end{figure}
            \end{example}
            \section{Background}
See the seminal scientific researches \cite{HG1,HG2,HG3}. The formalization of the notions on the framework of notions in SuperHyperGraphs, Neutrosophic notions in SuperHyperGraphs theory, and (Neutrosophic) SuperHyperGraphs theory at \cite{HG4,HG5,HG6,HG7,HG8,HG9,HG10,HG11,HG12,HG13,HG14,HG15,HG16,HG17,HG18,HG19,HG20,HG21,HG38}. Two popular scientific research books in Scribd in the terms of high readers, 4216 and 5214 respectively,  on neutrosophic science is on \cite{HG32b,HG44b}.

\begin{thebibliography}{595}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\bibitem{HG18}  Henry Garrett, \textit{``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints''}, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).
\bibitem{HG19}  Henry Garrett, \textit{``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances''}, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).
 \bibitem{HG20} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses}'', Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1).
\bibitem{HG21} Henry Garrett, ``\textit{SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions}'', Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1).

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

\end{thebibliography}
\end{document}