\documentclass[10pt,letterpaper]{article} \usepackage[top=0.85in,left=2.79in,footskip=0.79in,marginparwidth=2in]{geometry} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, tikz, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} % use Unicode characters - try changing the option if you run into troubles with special characters (e.g. umlauts) \usepackage[utf8]{inputenc} % clean citations \usepackage{cite} % hyperref makes references clicky. use \url{www.example.com} or \href{www.example.com}{description} to add a clicky url \usepackage{nameref,hyperref} % line numbers \usepackage[right]{lineno} % improves typesetting in LaTeX \usepackage{microtype} \DisableLigatures[f]{encoding = *, family = * } % text layout - change as needed \raggedright \setlength{\parindent}{0.5cm} \textwidth 5.25in \textheight 8.79in % Remove % for double line spacing %\usepackage{setspace} %\doublespacing % use adjustwidth environment to exceed text width (see examples in text) \usepackage{changepage} % adjust caption style \usepackage[aboveskip=1pt,labelfont=bf,labelsep=period,singlelinecheck=off]{caption} % remove brackets from references \makeatletter \renewcommand{\@biblabel}[1]{\quad#1.} \makeatother % headrule, footrule and page numbers \usepackage{lastpage,fancyhdr,graphicx} \usepackage{epstopdf} \pagestyle{myheadings} \pagestyle{fancy} \fancyhf{} \rfoot{\thepage/\pageref{LastPage}} \renewcommand{\footrule}{\hrule height 2pt \vspace{2mm}} \fancyheadoffset[L]{2.25in} \fancyfootoffset[L]{2.25in} % use \textcolor{color}{text} for colored text (e.g. highlight to-do areas) \usepackage{color} % define custom colors (this one is for figure captions) \definecolor{Gray}{gray}{.25} % this is required to include graphics \usepackage{graphicx} % use if you want to put caption to the side of the figure - see example in text \usepackage{sidecap} \usepackage{leftidx} % use for have text wrap around figures \usepackage{wrapfig} \usepackage[pscoord]{eso-pic} \usepackage[fulladjust]{marginnote} \reversemarginpar \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \theoremstyle{observation} \newtheorem{observation}[theorem]{Observation} \theoremstyle{question} \newtheorem{question}[theorem]{Question} \theoremstyle{problem} \newtheorem{problem}[theorem]{Problem} \numberwithin{equation}{section} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \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 } % document begins here \begin{document} \vspace*{0.35in} \linenumbers % title goes here: \begin{flushleft} {\Large \textbf\newline{ New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph } } \newline \newline Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA % authors go here: \end{flushleft} \section{ABSTRACT} In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyperHamiltonian-Cycle-Cut). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a Hamiltonian-Cycle-Cut 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-SuperHyperHamiltonian-Cycle-Cut if the following expression is called Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut criteria holds \begin{eqnarray*} && \forall E_a\in C: C~\text{is} \\&& \text{a SuperHyperCycle and it has} \\&& \text{the maximum number of SuperHyperVertices}; \end{eqnarray*} Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut if the following expression is called Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut criteria holds \begin{eqnarray*} && \forall E_a\in C: C~\text{is} \\&& \text{a SuperHyperCycle and it has} \\&& \text{the maximum number of SuperHyperVertices}; \end{eqnarray*} and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut if the following expression is called Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut criteria holds \begin{eqnarray*} && \forall V_a\in C: C~\text{is} \\&& \text{a SuperHyperCycle and it has} \\&& \text{the maximum number of SuperHyperVertices}; \end{eqnarray*} Neutrosophic rv-SuperHyperHamiltonian-Cycle-Cut if the following expression is called Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut criteria holds \begin{eqnarray*} && \forall V_a\in C: C~\text{is} \\&& \text{a SuperHyperCycle and it has} \\&& \text{the maximum number of SuperHyperVertices}; \end{eqnarray*} and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyperHamiltonian-Cycle-Cut if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-Cut. ((Neutrosophic) SuperHyperHamiltonian-Cycle-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 an Extreme SuperHyperHamiltonian-Cycle-Cut if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; a Neutrosophic SuperHyperHamiltonian-Cycle-Cut if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; an Extreme SuperHyperHamiltonian-Cycle-Cut SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperHamiltonian-Cycle-Cut SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyperHamiltonian-Cycle-Cut if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; a Neutrosophic V-SuperHyperHamiltonian-Cycle-Cut if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; an Extreme V-SuperHyperHamiltonian-Cycle-Cut SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperHamiltonian-Cycle-Cut SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic re-SuperHyperHamiltonian-Cycle-Cut, Neutrosophic v-SuperHyperHamiltonian-Cycle-Cut, and Neutrosophic rv-SuperHyperHamiltonian-Cycle-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 SuperHyperHamiltonian-Cycle-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyperHamiltonian-Cycle-Cut and Neutrosophic SuperHyperHamiltonian-Cycle-Cut. Two different types of SuperHyperDefinitions are debut for them but the research goes further and the SuperHyperNotion, SuperHyperUniform, and SuperHyperClass based on that are well-defined and well-reviewed. The literature review is implemented in the whole of this research. For shining the elegancy and the significancy of this research, the comparison between this SuperHyperNotion with other SuperHyperNotions and fundamental SuperHyperNumbers are featured. The definitions are followed by the examples and the instances thus the clarifications are driven with different tools. The applications are figured out to make sense about the theoretical aspect of this ongoing research. The ``Cancer's Recognition'' are the under research to figure out the challenges make sense about ongoing and upcoming research. The special case is up. The cells are viewed in the deemed ways. There are different types of them. Some of them are individuals and some of them are well-modeled by the group of cells. These types are all officially called ``SuperHyperVertex'' but the relations amid them all officially called ``SuperHyperEdge''. The frameworks ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph'' are chosen and elected to research about ``Cancer's Recognition''. Thus these complex and dense SuperHyperModels open up some avenues to research on theoretical segments and ``Cancer's Recognition''. Some avenues are posed to pursue this research. It's also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph. Assume a SuperHyperGraph. Then $\delta-$SuperHyperHamiltonian-Cycle-Cut is a maximal of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (Neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$ there are $|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta;$ and $ |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$ The first Expression, holds if $S$ is an $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an $\delta-$SuperHyperDefensive; a Neutrosophic $\delta-$SuperHyperHamiltonian-Cycle-Cut is a maximal Neutrosophic of SuperHyperVertices with maximum Neutrosophic cardinality such that either of the following expressions hold for the Neutrosophic cardinalities of SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{Neutrosophic} > |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta;$ and $ |S\cap N(s)|_{Neutrosophic} < |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta.$ The first Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperDefensive It's useful to define a ``Neutrosophic'' version of a SuperHyperHamiltonian-Cycle-Cut . Since there's more ways to get type-results to make a SuperHyperHamiltonian-Cycle-Cut more understandable. For the sake of having Neutrosophic SuperHyperHamiltonian-Cycle-Cut, there's a need to ``redefine'' the notion of a ``SuperHyperHamiltonian-Cycle-Cut ''. 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 SuperHyperHamiltonian-Cycle-Cut . It's redefined a Neutrosophic SuperHyperHamiltonian-Cycle-Cut if the mentioned Table holds, concerning, ``The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph'' with the key points, ``The Values of The Vertices \& The Number of Position in Alphabet'', ``The Values of The SuperVertices\&The maximum Values of Its Vertices'', ``The Values of The Edges\&The maximum Values of Its Vertices'', ``The Values of The HyperEdges\&The maximum Values of Its Vertices'', ``The Values of The SuperHyperEdges\&The maximum Values of Its Endpoints''. To get structural examples and instances, I'm going to introduce the next SuperHyperClass of SuperHyperGraph based on a SuperHyperHamiltonian-Cycle-Cut . 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 SuperHyperHamiltonian-Cycle-Cut until the SuperHyperHamiltonian-Cycle-Cut, then it's officially called a ``SuperHyperHamiltonian-Cycle-Cut'' but otherwise, it isn't a SuperHyperHamiltonian-Cycle-Cut . There are some instances about the clarifications for the main definition titled a ``SuperHyperHamiltonian-Cycle-Cut ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyperHamiltonian-Cycle-Cut . For the sake of having a Neutrosophic SuperHyperHamiltonian-Cycle-Cut, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyperHamiltonian-Cycle-Cut'' and a ``Neutrosophic SuperHyperHamiltonian-Cycle-Cut ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a Neutrosophic SuperHyperGraph. It's redefined ``Neutrosophic SuperHyperGraph'' if the intended Table holds. And a SuperHyperHamiltonian-Cycle-Cut are redefined to a ``Neutrosophic SuperHyperHamiltonian-Cycle-Cut'' if the intended Table holds. It's useful to define ``Neutrosophic'' version of SuperHyperClasses. Since there's more ways to get Neutrosophic type-results to make a Neutrosophic SuperHyperHamiltonian-Cycle-Cut more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHyperHamiltonian-Cycle-Cut, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are ``Neutrosophic SuperHyperPath'', ``Neutrosophic SuperHyperHamiltonian-Cycle-Cut'', ``Neutrosophic SuperHyperStar'', ``Neutrosophic SuperHyperBipartite'', ``Neutrosophic SuperHyperMultiPartite'', and ``Neutrosophic SuperHyperWheel'' if the intended Table holds. A SuperHyperGraph has a ``Neutrosophic SuperHyperHamiltonian-Cycle-Cut'' where it's the strongest [the maximum Neutrosophic value from all the SuperHyperHamiltonian-Cycle-Cut amid the maximum value amid all SuperHyperVertices from a SuperHyperHamiltonian-Cycle-Cut .] SuperHyperHamiltonian-Cycle-Cut . A graph is a SuperHyperUniform if it's a SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume a Neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows. It's SuperHyperPath if it's only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions; it's SuperHyperHamiltonian-Cycle-Cut 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