New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Bipartite and Path By Triangle-free search

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[HG371] Henry Garrett, "New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph With Bipartite and Path By Triangle-free search", Zenodo 2024, (doi: 10.5281/zenodo.13826744).

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\section{SYNOPSIS}
In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyper{\tiny Triangle-free search}). Let a Neutrosophic SuperHyperGraph (SHG) $S$ be a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet ${E_s}^{'}=\{E_1,E_2,\ldots,E_s\}.$ Then ${E_s}^{'}$ is called Neutrosophic SuperHyper{\tiny Triangle-free search} if the following expressions, terms and statements is called Neutrosophic SuperHyper{\tiny Triangle-free search} criteria holds. 
  \begin{eqnarray*}
&&|(\sum_{V_i\in {V_s}^{'}}T(V_i),\sum_{V_i\in {V_s}^{'}}I(V_i),\sum_{V_i\in {V_s}^{'}}F(V_i))|
\\&&
=\max_{{V_s}^{'}'\subseteq V_{NSHG}} |\{~(\sum_{V_i\in {V_s}^{'}'}T(V_i),\sum_{V_i\in {V_s}^{'}'}I(V_i),\sum_{V_i\in {V_s}^{'}'}F(V_i))~|
\\&&
~ \forall V_i,V_j,V_k,V_l\subseteq {V_s}^{'}':
\\&&
E^{STRONG}_{V_i,V_j,V_k,V_l}=\{V_iV_j,V_jV_k,V_kV_l,V_lV_i\}
\\&&
\}|.
\end{eqnarray*}
In this scientific research, new configurations are introduced for the new SuperHyperNotions, namely a Triangle-free search and a Neutrosophic Triangle-free search. 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 Triangle-free search. Since there are more ways to get type results a Triangle-free search becomes more understandable. In order to master the Neutrosophic Triangle-free search, there is a need to ``re-define" the concept of ``Triangle-free search mastery". SuperHyperVertices and SuperHyperEdges are assigned by alphabetical labels. In this method, the position of tags is used to assign values. Suppose a Triangle-free search. 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 Triangle-free search. 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 Triangle-free search until Triangle-free search, then it is officially called ``Triangle-free search", but otherwise, it is not Triangle-free search. There is some explanation for the original definition titled ``Triangle-free search".  These two examples are further explored and distinguished because they are characterized in the SuperHyperClass disciplinary methods based on a dominant Triangle-free search. Because of having Triangle-free search, there is a need to ``re-define" the concept of ``neutrosophic Triangle-free search" and ``neutrosophic Triangle-free search". 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 Triangle-free search is redefined to a ``Neutrosophic Triangle-free search". It is useful to define a ``Neutrosophic" version of SuperHyperClasses. Since there are more ways to obtain Neutrosophic type results, a SuperHyper- Neutrosophic Triangle-free search 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 Triangle-free search or the strongest Triangle-free search in those Neutrosophic SuperHyperModels. Some general results are presented for the longest Triangle-free search called Triangle-free search and the strongest Triangle-free search called Neutrosophic Triangle-free search. 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 Triangle-free search Neutrosophic, SuperHyperGraphs and Neutrosophic SuperHyperGraphs is suggested.
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\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyper{\tiny Triangle-free search}, Cancer's Neutrosophic Recognition 
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\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45