New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Bipartite and Path By EQUAL CONNECTIVE DOMINATION As Hyper Decoration On Super Density

307 [HG307] Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Bipartite and Path By EQUAL CONNECTIVE DOMINATION As Hyper Decoration On Super Density”, Zenodo 2023, (doi: 10.5281/zenodo.8201255). @ResearchGate: https://www.researchgate.net/publication/- @Scribd: https://www.scribd.com/document/- @ZENODO_ORG: https://zenodo.org/record/8201255 @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 In Recognition of Cancer And Neutrosophic SuperHyperGraph With Bipartite and Path By EQUAL CONNECTIVE DOMINATION As Hyper Decoration On Super Density } } \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 EQUAL CONNECTIVE DOMINATION}). Let a Neutrosophic SuperHyperGraph (NSHG) $S$ be a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}.$ Then $V'$ is called Neutrosophic SuperHyper{\tiny EQUAL CONNECTIVE DOMINATION} if the following expressions, terms and statements is called Neutrosophic SuperHyper{\tiny EQUAL CONNECTIVE DOMINATION} criteria holds. \begin{eqnarray*} &&|(\sum_{V_i\in V'}T(V_i),\sum_{V_i\in V'}I(V_i),\sum_{V_i\in V'}F(V_i))| \\&& =\max_{V''\subseteq V_{NSHG}} |\{~(\sum_{V_i\in V''}T(V_i),\sum_{V_i\in V''}I(V_i),\sum_{V_i\in V''}F(V_i))~|~\forall V_i\in V_{NSHG},\exists V_j\in V'': \\&&T'_V(V_i,V_j\in E_{i'})= \min[T_{V}(V'_i),T_{V}(V'_j)]_{V'_i,V'_j\in P:V_i,\ldots,V_j}, \\&& I'_V(V_i,V_j\in E_{i'})=\min[I_{V}(V'_i),I_{V}(V'_j)]_{V_i,V_j\in E_{i'}}_{V'_i,V'_j\in P:V_i,\ldots,V_j}, \\&& \text{and}~F'_V(E_{i'})(V_i,V_j\in E_{i'})=\min[F_{V}(V'_i),F_{V}(V'_j)]_{V_i,V_j\in E_{i'}}_{V'_i,V'_j\in P:V_i,\ldots,V_j} \\&& \} \end{eqnarray*} In this scientific research, new configurations are introduced for the new SuperHyperNotions, namely a EQUAL CONNECTIVE DOMINATION and a Neutrosophic EQUAL CONNECTIVE DOMINATION. Two different SuperHyperDefinitions were started for them, but research has progressed and SuperHyperNotion, SuperHyperUniform, and SuperHyperClass are well-defined and well-reviewed based on them. Literature review is implemented throughout this research. To shine the light and importance of this research, a comparison between this SuperHyperNotion with other SuperHyperNotions and basic SuperHyperNumbers is presented. Definitions are followed by examples and examples, so clarifications are guided by various tools. Applications are designed to understand the theoretical aspect of this ongoing research. ``Cancer diagnosis" are the things that are being researched to explore the challenges that make sense in ongoing research and future research. It is a special case. Cells are viewed in the intended manner. There are different types of them. Some are individual and some are well modeled by groups of cells. All these types are officially called ``SuperHyperVertex", but the relationships between them all are officially called ``SuperHyperEdge". ``SuperHyperGraph" and ``Neutrosophic SuperHyperGraph" frameworks are selected and selected for research on ``Cancer Diagnosis". Thus, these complex and dense super-hypermodels open avenues for research on theoretical aspects and ``cancer diagnosis". Ways to pursue this scientific research have been considered. It has also been formally collected in the form of several questions and problems. It is useful to define a ``neutrosophic" version of EQUAL CONNECTIVE DOMINATION. Since there are more ways to get type results a EQUAL CONNECTIVE DOMINATION becomes more understandable. In order to master the Neutrosophic EQUAL CONNECTIVE DOMINATION, there is a need to ``re-define" the concept of ``EQUAL CONNECTIVE DOMINATION mastery". SuperHyperVertices and SuperHyperEdges are assigned by alphabetical labels. In this method, the position of tags is used to assign values. Suppose a EQUAL CONNECTIVE DOMINATION. If the specified table contains ``values of vertices, supervertices, edges, hyperedges, and hyperedges belonging to Neutrosophic SuperHyperGraph" with key points, it is redefined. "values of vertices \& number of positions in alphabet", "Values of SuperVertices\&maximum values of its vertices", "values of edges\&maximum values of its vertices", "super edge values\&max values of its vertices", ``super edge values" and maximum values of its endpoints". To get examples and structural examples, I would like to introduce the next SuperHyperClass SuperHyperGraph based on EQUAL CONNECTIVE DOMINATION. This is the original. Having the foundation of the previous definition in the SuperHyperClass type would be disciplined. If there is a need to have all EQUAL CONNECTIVE DOMINATION until EQUAL CONNECTIVE DOMINATION, then it is officially called ``EQUAL CONNECTIVE DOMINATION", but otherwise, it is not EQUAL CONNECTIVE DOMINATION. There is some explanation for the original definition titled ``EQUAL CONNECTIVE DOMINATION". These two examples are further explored and distinguished because they are characterized in the SuperHyperClass disciplinary methods based on a dominant EQUAL CONNECTIVE DOMINATION. Because of having EQUAL CONNECTIVE DOMINATION, there is a need to ``re-define" the concept of ``neutrosophic EQUAL CONNECTIVE DOMINATION" and ``neutrosophic EQUAL CONNECTIVE DOMINATION". SuperHyperVertices and SuperHyperEdges are assigned by alphabetical labels. In this method, the position of tags is used to assign values. Consider a Neutrosophic SuperHyperGraph. ``Neutrosophic SuperHyperGraph" is redefined if the desired table exists. And if the desired table exists, a EQUAL CONNECTIVE DOMINATION is redefined to a ``Neutrosophic EQUAL CONNECTIVE DOMINATION". It is useful to define a ``Neutrosophic" version of SuperHyperClasses. Since there are more ways to obtain Neutrosophic type results, a SuperHyper- Neutrosophic EQUAL CONNECTIVE DOMINATION becomes more understandable. Consider a Neutrosophic SuperHyperGraph. If the desired table exists, there are a number of Neutrosophic SuperHyperClasses. So are SuperHyperPath, SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite and SuperHyperWheel. "Neutrosophic SuperHyperPath", ``Neutrosophic SuperHyperCycle", ``Neutrosophic SuperHyperStar", ``Neutrosophic SuperHyperBipartite", ``Neutrosophic SuperHyperMultiPartite" and ``Neutrosophic SuperHyperWheel". A graph is SuperHyperUniform if it is a SuperHyperGraph and the number of SuperHyperEdges elements is the same. Consider a Neutrosophic SuperHyperGraph. There are a number of SuperHyperClasses as follows. It is a SuperHyperPath if there is only one SuperVertex as an intersection between two given SuperHyperEdges with two exceptions. If only one SuperVertex is given as an intersection between two SuperHyperEdges, it is SuperHyperCycle. It is SuperHyperStar and only one SuperVertex as intersection among all SuperHyperEdges. It is SuperHyperBipartite, and only one SuperVertex is given as the intersection between two SuperHyperEdges, and these SuperVertices, which form two distinct sets, have no SuperHyperEdge in common. This is SuperHyperMultiPartite. Only one SuperVertex is given as an intersection between two SuperHyperEdges, and these SuperVertices, which form separate multisets, have no SuperHyperEdge in common. If only one SuperHyperEdge among the given two SuperHyperEdges is a SuperHyperWheel and a SuperVertex has a SuperHyperEdge with any regular SuperVertex. SuperHyperModel suggests specific designs and specific architectures. SuperHyperModel is officially called ``SuperHyperGraph" and ``Neutrosophic SuperHyperGraph". In this SuperHyperModel, the ``special" cells and the ``special group" of cells are SuperHyperModeled as ``SuperHyperVertices" and the common and desired properties between the ``special" cells and the ``special group" of cells. SuperHyperModeled as ``SuperHyperEdges". Sometimes, to have a more accurate SuperHyperModel, it is useful to have some degree of determinism, uncertainty, and neutrality, in this case the SuperHyperModel is called ``Neutrosophic". In future scientific research, the foundation will be based on ``diagnosis of cancer" and the results and definitions will be introduced in redeemed methods. Cancer diagnosis in long-term practice. The specific region is assigned by the model (called SuperHyperGraph) and the long cycle of movement of the cancer is determined by this scientific research. Sometimes the movement of cancer is not easy to detect, because there is certainty, uncertainty and neutrality about the movements and effects of cancer on that area. This event leads us to choose another model [which is said to be the Neutrosophic superhypergraph] in order to have a proper understanding of what happened and was done. There are some specific models, which are well known and have names, and some SuperHyperGeneral SuperHyperModels. The movements and effects of cancer in complex pathways and between complex groups of cells can be visualized with a Neutrosophic SuperHyperPath (-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The goal is to find the longest EQUAL CONNECTIVE DOMINATION or the strongest EQUAL CONNECTIVE DOMINATION in those Neutrosophic SuperHyperModels. Some general results are presented for the longest EQUAL CONNECTIVE DOMINATION called EQUAL CONNECTIVE DOMINATION and the strongest EQUAL CONNECTIVE DOMINATION called Neutrosophic EQUAL CONNECTIVE DOMINATION. Beyond that in SuperHyperStar, all possible SuperHyperPaths have only two SuperHyperEdges, but it is not enough as having at least three SuperHyperEdges is necessary to form any style of cycle. There is no form of any cycle, but literally, it is the transformation of any cycle. It literally transforms and does not form. A basic introduction to the theory of EQUAL CONNECTIVE DOMINATION Neutrosophic, SuperHyperGraphs and Neutrosophic SuperHyperGraphs is suggested. \\ \vspace{4mm} \textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyper{\tiny EQUAL CONNECTIVE DOMINATION}, Cancer's Neutrosophic Recognition \\ \textbf{AMS Subject Classification:} 05C17, 05C22, 05E45 \section{Applied Notions Under The Scrutiny Of The Motivation Of This Scientific Research} In this scientific research, there are ideas in the prominent frameworks of motivations. I try to bring motivations in narrative ways. Some Cells have faced attacks from the established situation with cancer attacks in this case, there is some analysis embedded in the case current situations where cells can be labeled as some groups and some groups or individuals are overlabeled, all from behaviors to overcome cancer attacks. In embedded situations, cell individuals and cell groups can be considered as ``new groups". So it encourages us to find suitable SuperHyperModels for more appropriate analysis about this dirty story. I found SuperHyperModels, officially called ``SuperHyperGraphs" and ``Neutrosophic SuperHyperGraphs". In this SuperHyperModel, cells and groups of cells are defined as ``SuperHyperVertices" and relationships between individual cells and groups of cells are defined as ``SuperHyperEdges". So this is another motivation for us to do research on this SuperHyperModel based on ``cancer diagnosis". Sometimes, things get worse. The situation has passed certainty and precise style. So it is beyond them. There are three descriptions, i.e., degrees of determination, uncertainty, and neutrality, for any object based on ambiguous forms, i.e., incomplete data, imprecise data, and uncertain analysis. The second model can be considered in the previous SuperHyperModel. This is SuperHyperModel. This is SuperHyperGraph but officially called ``Neutrosophic SuperHyperGraphs". Cancer is a disease, but the model is supposed to understand what happens in this phenomenon. The special case of this disease is considered and some parameters are used as consequences of the model. Cells are attacked by the disease, but the movement of cancer is in a specific area. Knowing the mind of cancer can help to find some cures for this disease. SuperHyperGraph and Neutrosophic Super-HyperGraph are super-hypermodels used in ``cancer diagnosis" and both are the foundation of the background of this scientific research. Sometimes cancer has occurred in an area full of cells, cell groups, and embedded styles. In this section, SuperHyperModel proposes based on the links of cancer movements in the form of unity styles with the formation of design and architecture officially called ``EQUAL CONNECTIVE DOMINATION", in the themes of specialized terms and common words. Particle for direct object. Applied concepts under the investigation of the motivation of this scientific research. The ``SuperHyper" prefix refers to the theme of embedded styles for background discovery for SuperHyperNotions. Cancer diagnosis in long-term practice. The specific region is assigned by the model (called SuperHyperGraph) and the long cycle of movement of the cancer is determined by this scientific research. Sometimes the movement of Cancer is not easy to identify because there is certainty, uncertainty and neutrality about the movements and effects of Cancer in that region. This event leads us to choose another model [said to be Neutrosophic SuperHyperGraph]. Easy understanding of what happened and what was done. There are some specific models that are well known and named and some general models. The movements and effects of cancer in complex pathways and between complex groups of cells can be visualized by a neurosophic superhyperpath (-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The goal is to find EQUAL CONNECTIVE DOMINATION or Neutrosophic EQUAL CONNECTIVE DOMINATION for SuperHyperModels in it. Some general results are presented. Beyond that in SuperHyperStar, all possible Neutrosophic SuperHyperPaths have only two SuperHyperEdges, but it is not enough as having at least three SuperHyperEdges is necessary to form any style of cycle. There is no form of cycle, but it is literally a form of any cycle. It literally transforms and does not form. \begin{question} How to define ``SuperHyperNotions and to do research" to find the ``EQUAL CONNECTIVE DOMINATION Rate" of any cell or groups of cells based on fixed cell or fixed group of cells broadly, ``EQUAL CONNECTIVE DOMINATION Rate" based on fixed groups or fixed cells. Groups of groups of cells? \end{question} \begin{question} What are the best descriptions for ``diagnosing cancer" in terms of these chaotic and dense super-hypermodels in which embedded concepts are depicted? \end{question} This motivation to find concepts to use in this dense model is called ``SuperHyperGraphs". So it motivates us to define different types of ``EQUAL CONNECTIVE DOMINATION" and ``neutrosophic EQUAL CONNECTIVE DOMINATION" over ``superhypergraph" and ``neutrosophic superhypergraph". Then this scientific research has taken more motivations to define SuperHyperClasses and find some connections among this SuperHyperNotion with other SuperHyperNotions. It motivates us to obtain examples and examples to clarify the framework of this scientific research. The general results and some of the results about some of the connections are the ways that the key point of this scientific research, ``diagnosis of cancer" is more understandable and clear. \\ The framework of this research is as follows. First, I will introduce the basic definitions to clarify the basics. In the ``Introductions" subsection, basic definitions about SuperHyperGraphs and Neutrosophic SuperHyperGraph are introduced and discussed in depth. Basic concepts are thoroughly explained and illustrated, and sometimes a literature review is used to make sense of what is to come in the coming sections. The main definitions and their explanations, along with some results on the new concepts, EQUAL CONNECTIVE DOMINATION and Neutrosophic EQUAL CONNECTIVE DOMINATION, are specified in the sections ``EQUAL CONNECTIVE DOMINATION" and ``Neutrosophic EQUAL CONNECTIVE DOMINATION". In the concept of dealing with results and in EQUAL CONNECTIVE DOMINATION to create logic about the continuation of the scientific research, the ideas of SuperHyperUniform and Neutrosophic SuperHyperUniform are introduced and as their consequences, the corresponding SuperHyperClasses are specified to summarize the work done in this section, as ``Results on SuperHyperClasses" and ``Results on Neutrosophic SuperHyperClasses". Going back to the core concepts, clever steps towards common concepts to extend new concepts in new frameworks, SuperHyperGraph and Neutrosophic SuperHyperGraph, in ``Results on SuperHyperClasses" and ``Results on Neutrosophic SuperHyperClasses" sections. There is the initial scientific research on supergeneral relations and as the final and final part of the theoretical scientific research is available in the section ``General results". Some general SuperHyperRelations are fundamental and are known as fundamental SuperHyperNotions, as in ``General results" were extracted and discussed. ``EQUAL CONNECTIVE DOMINATION", ``Neutrosophic EQUAL CONNECTIVE DOMINATION", ``Results on SuperHyperClasses" and ``Results on Neutrosophic SuperHyperClasses". There are curious questions about what has been done about SuperHyperNotions to make sense of the superiority of this scientific research and to discover the word ``best" as a description and adjective of this scientific research as presented in the ``EQUAL CONNECTIVE DOMINATION" section. The keyword of this scientific research was presented in the section ``Applications in cancer diagnosis" with two cases and sub-sections ``Case 1: Initial steps towards SuperHyperBipartite as SuperHyperModel" and ``Case 2: Incremental steps towards SuperHyperMultiperdel as SuperHyper. In the ``open issues" section, there is a detailed examination and recognition of what was done in this scientific research and what happened in this scientific research in the form of ``questions" and ``issues" in order to understand this scientific research in an outstanding style, the advantages and limitations of this scientific research in addition to What has been done in this scientific research is stated in the ``conclusion and final remarks" section. \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{HG11s},Definition 2.2,p.2), [Neutrosophic Set](\textbf{Ref.}\cite{HG11s},Definition 2.1,p.1), [N