New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Star and Wheel By Triangle-free search Edge

375 [HG375] Henry Garrett, "New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Star and Wheel By Triangle-free search Edge", Zenodo 2024, (doi: 10.5281/zenodo.13909326). @ResearchGate: https://www.researchgate.net/publication/- @Scribd: https://www.scribd.com/document/- @ZENODO_ORG: https://zenodo.org/record/13909326 @academia: https://www.academia.edu/- Dr. Henry Garrett · Independent Researcher · Department of Mathematics · [email protected] · Manhattan, NY, USA \section{SYNOPSIS}In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyper{\tiny Triangle-free search Edge}). Let a Neutrosophic SuperHyperGraph (SHG) $S$ be a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $A_s=\{E_1,E_2,\ldots,E_s\}.$ Then $A_s$ is called Neutrosophic SuperHyper{\tiny Triangle-free search Edge} if the following expressions, terms and statements is called Neutrosophic SuperHyper{\tiny Triangle-free search Edge} criteria holds. \begin{eqnarray*}&&|(\sum_{E_i\in A_s}T(E_i),\sum_{E_i\in A_s}I(E_i),\sum_{E_i\in A_s}F(E_i))|\\&&=\max_{B_s\subseteq E_{NSHG}} |\{~(\sum_{E_i\in B_s}T(E_i),\sum_{E_i\in B_s}I(E_i),\sum_{E_i\in B_s}F(E_i))~|\\&&~ \forall E_i,E_j,E_k\in B_s:\\&&V_a\in E_i,E_j~\&~V_b\in E_j,E_k\not\Rightarrow V_c\in E_i,E_k\\&&\}|.\end{eqnarray*}In this scientific research, new configurations are introduced for the new SuperHyperNotions, namely a Strongly Triangle-free search Edge and a Neutrosophic Strongly Triangle-free search Edge. 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 Strongly Triangle-free search Edge. Since there are more ways to get type results a Strongly Triangle-free search Edge becomes more understandable. In order to master the Neutrosophic Strongly Triangle-free search Edge, there is a need to ``re-define" the concept of ``Strongly Triangle-free search Edge mastery". SuperHyperVertices and SuperHyperEdges are assigned by alphabetical labels. In this method, the position of tags is used to assign values. Suppose a Strongly Triangle-free search Edge. 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 Strongly Triangle-free search Edge. 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 Strongly Triangle-free search Edge until Strongly Triangle-free search Edge, then it is officially called ``Strongly Triangle-free search Edge", but otherwise, it is not Strongly Triangle-free search Edge. There is some explanation for the original definition titled ``Strongly Triangle-free search Edge". These two examples are further explored and distinguished because they are characterized in the SuperHyperClass disciplinary methods based on a dominant Strongly Triangle-free search Edge. Because of having Strongly Triangle-free search Edge, there is a need to ``re-define" the concept of ``neutrosophic Strongly Triangle-free search Edge" and ``neutrosophic Strongly Triangle-free search Edge". 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 Strongly Triangle-free search Edge is redefined to a ``Neutrosophic Strongly Triangle-free search Edge". It is useful to define a ``Neutrosophic" version of SuperHyperClasses. Since there are more ways to obtain Neutrosophic type results, a SuperHyper- Neutrosophic Strongly Triangle-free search Edge 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 Strongly Triangle-free search Edge or the strongest Strongly Triangle-free search Edge in those Neutrosophic SuperHyperModels. Some general results are presented for the longest Strongly Triangle-free search Edge called Strongly Triangle-free search Edge and the strongest Strongly Triangle-free search Edge called Neutrosophic Strongly Triangle-free search Edge. 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 Strongly Triangle-free search Edge Neutrosophic, SuperHyperGraphs and Neutrosophic SuperHyperGraphs is suggested.\\ \vspace{4mm}\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyper{\tiny Strongly Triangle-free search Edge}, Cancer's Neutrosophic Recognition \\\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45