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298 [HG298] Henry Garrett, “New Ideas On Super Perfidy By Hyper Perf Of SuperHyperPerfect Search In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.8104857). @ResearchGate: https://www.researchgate.net/publication/- @Scribd: https://www.scribd.com/document/- @ZENODO_ORG: https://zenodo.org/record/8104857 @academia: https://www.academia.edu/- \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 Perfidy By Hyper Perf Of SuperHyperPerfect Search 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 SuperHyper{\tiny SuperHyperPerfect Search}). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a {\tiny SuperHyperPerfect Search} pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search} if the following expression is called Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search} criteria holds \begin{eqnarray*} && E' \text{has the maximum chromatic number and has the maximum clique number} \\&& \text{and its chromatic number is equal to its clique number;} \end{eqnarray*} Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search} if the following expression is called Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search} criteria holds \begin{eqnarray*} && E' \text{has the maximum chromatic number and has the maximum clique number} \\&& \text{and its chromatic number is equal to its clique number;} \end{eqnarray*} and $|E_i|_{\text{NEUTROSOPHIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPHIC CARDINALITY}};$ Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search} if the following expression is called Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search} criteria holds \begin{eqnarray*} && E' \text{has the maximum chromatic number and has the maximum clique number} \\&& \text{and its chromatic number is equal to its clique number;} \end{eqnarray*} Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} if the following expression is called Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search} criteria holds \begin{eqnarray*} && E' \text{has the maximum chromatic number and has the maximum clique number} \\&& \text{and its chromatic number is equal to its clique number;} \end{eqnarray*} and $|V_i|_{\text{NEUTROSOPHIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPHIC CARDINALITY}};$ Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search}. ((Neutrosophic) SuperHyper{\tiny SuperHyperPerfect Search}). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$ Then $E$ is called an Extreme SuperHyper{\tiny SuperHyperPerfect Search} if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; an Extreme SuperHyper{\tiny SuperHyperPerfect Search} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyper{\tiny SuperHyperPerfect Search} if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; a Neutrosophic V-SuperHyper{\tiny SuperHyperPerfect Search} if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; an Extreme V-SuperHyper{\tiny SuperHyperPerfect Search} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic re-SuperHyper{\tiny SuperHyperPerfect Search}, Neutrosophic v-SuperHyper{\tiny SuperHyperPerfect Search}, and Neutrosophic rv-SuperHyper{\tiny SuperHyperPerfect Search} 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 SuperHyper{\tiny SuperHyperPerfect Search}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyper{\tiny SuperHyperPerfect Search} and Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search}. Two different types of SuperHyperDefinitions are debut for them but the research goes further and the SuperHyperNotion, SuperHyperUniform, and SuperHyperClass based on that are well-defined and well-reviewed. The literature review is implemented in the whole of this research. For shining the elegancy and the significancy of this research, the comparison between this SuperHyperNotion with other SuperHyperNotions and fundamental SuperHyperNumbers are featured. The definitions are followed by the examples and the instances thus the clarifications are driven with different tools. The applications are figured out to make sense about the theoretical aspect of this ongoing research. The ``Cancer's Recognition'' are the under research to figure out the challenges make sense about ongoing and upcoming research. The special case is up. The cells are viewed in the deemed ways. There are different types of them. Some of them are individuals and some of them are well-modeled by the group of cells. These types are all officially called ``SuperHyperVertex'' but the relations amid them all officially called ``SuperHyperEdge''. The frameworks ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph'' are chosen and elected to research about ``Cancer's Recognition''. Thus these complex and dense SuperHyperModels open up some avenues to research on theoretical segments and ``Cancer's Recognition''. Some avenues are posed to pursue this research. It's also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph. Assume a SuperHyperGraph. Then $\delta-$SuperHyper{\tiny SuperHyperPerfect Search} is a maximal of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (Neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$ there are $|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta;$ and $ |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$ The first Expression, holds if $S$ is an $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an $\delta-$SuperHyperDefensive; a Neutrosophic $\delta-$SuperHyper{\tiny SuperHyperPerfect Search} is a maximal Neutrosophic of SuperHyperVertices with maximum Neutrosophic cardinality such that either of the following expressions hold for the Neutrosophic cardinalities of SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{Neutrosophic} > |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta;$ and $ |S\cap N(s)|_{Neutrosophic} < |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta.$ The first Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is a Neutrosophic $\delta-$SuperHyperDefensive It's useful to define a ``Neutrosophic'' version of a SuperHyper{\tiny SuperHyperPerfect Search} . Since there's more ways to get type-results to make a SuperHyper{\tiny SuperHyperPerfect Search} more understandable. For the sake of having Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search}, there's a need to ``redefine'' the notion of a ``SuperHyper{\tiny SuperHyperPerfect Search} ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a SuperHyper{\tiny SuperHyperPerfect Search} . It's redefined a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} if the mentioned Table holds, concerning, ``The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph'' with the key points, ``The Values of The Vertices \& The Number of Position in Alphabet'', ``The Values of The SuperVertices\&The maximum Values of Its Vertices'', ``The Values of The Edges\&The maximum Values of Its Vertices'', ``The Values of The HyperEdges\&The maximum Values of Its Vertices'', ``The Values of The SuperHyperEdges\&The maximum Values of Its Endpoints''. To get structural examples and instances, I'm going to introduce the next SuperHyperClass of SuperHyperGraph based on a SuperHyper{\tiny SuperHyperPerfect Search} . It's the main. It'll be disciplinary to have the foundation of previous definition in the kind of SuperHyperClass. If there's a need to have all SuperHyper{\tiny SuperHyperPerfect Search} until the SuperHyper{\tiny SuperHyperPerfect Search}, then it's officially called a ``SuperHyper{\tiny SuperHyperPerfect Search}'' but otherwise, it isn't a SuperHyper{\tiny SuperHyperPerfect Search} . There are some instances about the clarifications for the main definition titled a ``SuperHyper{\tiny SuperHyperPerfect Search} ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyper{\tiny SuperHyperPerfect Search} . For the sake of having a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search}, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search}'' and a ``Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a Neutrosophic SuperHyperGraph. It's redefined ``Neutrosophic SuperHyperGraph'' if the intended Table holds. And a SuperHyper{\tiny SuperHyperPerfect Search} are redefined to a ``Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search}'' if the intended Table holds. It's useful to define ``Neutrosophic'' version of SuperHyperClasses. Since there's more ways to get Neutrosophic type-results to make a Neutrosophic SuperHyper{\tiny SuperHyperPerfect Search} more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended |
