\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 EULA By Hyper Eulogy Of Eulerian-Type-Path-Cut In Cancer's Recognition 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 Eulerian-Type-Path-Cut}). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a {\tiny Eulerian-Type-Path-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-SuperHyper{\tiny Eulerian-Type-Path-Cut} if the following expression is called Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut} criteria holds \begin{eqnarray*} && \forall E_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the maximum number of SuperHyperEdges}; \end{eqnarray*} Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut} if the following expression is called Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut} criteria holds \begin{eqnarray*} && \forall E_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the maximum number of SuperHyperEdges}; \end{eqnarray*} and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut} if the following expression is called Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut} criteria holds \begin{eqnarray*} && \forall V_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the maximum number of SuperHyperEdges}; \end{eqnarray*} Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Path-Cut} if the following expression is called Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut} criteria holds \begin{eqnarray*} && \forall V_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the maximum number of SuperHyperEdges}; \end{eqnarray*} and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Path-Cut}. ((Neutrosophic) SuperHyper{\tiny Eulerian-Type-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 an Extreme SuperHyper{\tiny Eulerian-Type-Path-Cut} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; an Extreme SuperHyper{\tiny Eulerian-Type-Path-Cut} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyper{\tiny Eulerian-Type-Path-Cut} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; a Neutrosophic V-SuperHyper{\tiny Eulerian-Type-Path-Cut} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; an Extreme V-SuperHyper{\tiny Eulerian-Type-Path-Cut} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Path-Cut}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Path-Cut}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-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 SuperHyper{\tiny Eulerian-Type-Path-Cut}; 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 Eulerian-Type-Path-Cut} and Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-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 $\delta-$SuperHyper{\tiny Eulerian-Type-Path-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 $\delta-$SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-Cut} . Since there's more ways to get type-results to make a SuperHyper{\tiny Eulerian-Type-Path-Cut} more understandable. For the sake of having Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut}, there's a need to ``redefine'' the notion of a ``SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-Cut} . It's redefined a Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-Cut} until the SuperHyper{\tiny Eulerian-Type-Path-Cut}, then it's officially called a ``SuperHyper{\tiny Eulerian-Type-Path-Cut}'' but otherwise, it isn't a SuperHyper{\tiny Eulerian-Type-Path-Cut} . There are some instances about the clarifications for the main definition titled a ``SuperHyper{\tiny Eulerian-Type-Path-Cut} ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyper{\tiny Eulerian-Type-Path-Cut} . For the sake of having a Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut}, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut}'' and a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-Cut} are redefined to a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-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 SuperHyper{\tiny Eulerian-Type-Path-Cut} more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHyper{\tiny Eulerian-Type-Path-Cut}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are ``Neutrosophic SuperHyperPath'', ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut}'', ``Neutrosophic SuperHyperStar'', ``Neutrosophic SuperHyperBipartite'', ``Neutrosophic SuperHyperMultiPartite'', and ``Neutrosophic SuperHyperWheel'' if the intended Table holds. A SuperHyperGraph has a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Path-Cut}'' where it's the strongest [the maximum Neutrosoph