(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses
Authors: Garrett Henry 
Published: December 2022

In this research, new setting is introduced for new SuperHyperNotions, namely, SuperHyperAlliances and Neutrosophic SuperHyperAlliances. 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 &ldquo;Cancer&rsquo;s Recognitions&rdquo; 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 &ldquo;SuperHyperVertex&rdquo; but the relations amid them all officially called &ldquo;SuperHyperEdge&rdquo;. The frameworks &ldquo;SuperHyperGraph&rdquo; and &ldquo;neutrosophic SuperHyperGraph&rdquo; are chosen and elected to research about &ldquo;Cancer&rsquo;s Recognitions&rdquo;. Thus these complex and dense SuperHyperModels open up some avenues to research on theoretical segments and &ldquo;Cancer&rsquo;s Recognitions&rdquo;. Some avenues are posed to pursue this research. It&rsquo;s also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph. An &ldquo;SuperHyperAlliance&rdquo; is a minimal SuperHyperSet of SuperHyperVertices with minimum cardinality such that either of the following expressions hold for the cardinalities of SuperHyperNeighbors of s&isin;S:,|S&cap;N(s)|&gt;|S&cap;(V \N(s))|,and|S&cap;N(s)|&lt;|S&cap;(V \N(s))|.Thefirst Expression, holds if S is SuperHyperOffensive. And the second Expression, holds if S is &ldquo;SuperHyperDefensive&rdquo;. It&rsquo;s useful to define &ldquo;neutrosophic&rdquo; version of SuperHyperAlliances. Since there&rsquo;s more ways to get type-results to make SuperHyperAlliances more understandable. For the sake of having neutrosophic SuperHyperAlliances, there&rsquo;s a need to &ldquo;redefine&rdquo; the notion of &ldquo;SuperHyperAlliances&rdquo;. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there&rsquo;s the usage of the position of labels to assign to the values. Assume a SuperHyperAlliance. It&rsquo;s redefined neutrosophic SuperHyperAlliance if the mentioned Table holds, concerning, &ldquo;The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph&rdquo; with the key points, &ldquo;The Values of The Vertices &amp; The Number of Position in Alphabet&rdquo;, &ldquo;The Values of The SuperVertices&amp;The Minimum Values of Its Vertices&rdquo;, &ldquo;The Values of The Edges&amp;The Minimum Values of Its Vertices&rdquo;, &ldquo;The Values of The HyperEdges&amp;The Minimum Values of Its Vertices&rdquo;, &ldquo;The Values of The SuperHyperEdges&amp;The Minimum Values of Its Endpoints&rdquo;. To get structural examples and instances, I&rsquo;m going to introduce the next SuperHyperClass of SuperHyperGraph based on SuperHyperAlliances. It&rsquo;s the main. It&rsquo;ll be disciplinary to have the foundation of previous definition in the kind of SuperHyperClass. If there&rsquo;s a need to have all SuperHyperConnectivities until the SuperHyperAlliances, then it&rsquo;s officially called &ldquo;SuperHyperAlliances&rdquo; but otherwise, it isn&rsquo;t SuperHyperAlliances. There are some instances about the clarifications for the main definition titled &ldquo;SuperHyperAlliances&rdquo;. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on SuperHyperAlliances. For the sake of having neutrosophic SuperHyperAlliances, there&rsquo;s a need to &ldquo;redefine&rdquo; the notion of &ldquo;neutrosophic SuperHyperAlliances&rdquo; and &ldquo;neutrosophic SuperHyperAlliances&rdquo;. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there&rsquo;s the usage of the position of labels to assign to the values. Assume a neutrosophic SuperHyperGraph. It&rsquo;s redefined &ldquo;neutrosophic SuperHyperGraph&rdquo; if the intended Table holds. And SuperHyperAlliances are redefined &ldquo;neutrosophic SuperHyperAlliances&rdquo; if the intended Table holds. It&rsquo;s useful to define &ldquo;neutrosophic&rdquo; version of SuperHyperClasses. Since there&rsquo;s more ways to get neutrosophic type-results to make neutrosophic SuperHyperAlliances more understandable. Assume a neutrosophic SuperHyperGraph. There are some neutrosophic SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are &ldquo;neutrosophic SuperHyperPath&rdquo;, &ldquo;neutrosophic SuperHyperCycle&rdquo;, &ldquo;neutrosophic SuperHyperStar&rdquo;, &ldquo;neutrosophic SuperHyperBipartite&rdquo;, &ldquo;neutrosophic SuperHyperMultiPartite&rdquo;, and &ldquo;neutrosophic SuperHyperWheel&rdquo; if the intended Table holds. A SuperHyperGraph has &ldquo;neutrosophic SuperHyperAlliances&rdquo; where it&rsquo;s the strongest [the maximum neutrosophic value from all SuperHyperAlliances amid the maximum value amid all SuperHyperVertices from a SuperHyperAlliances.] SuperHyperAlliances. A graph is SuperHyperUniform if it&rsquo;s SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume a neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows. It&rsquo;s SuperHyperPath if it&rsquo;s only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions; it&rsquo;s SuperHyperCycle if it&rsquo;s only one SuperVertex as intersection amid two given SuperHyperEdges; it&rsquo;s SuperHyperStar it&rsquo;s only one SuperVertex as intersection amid all SuperHyperEdges; it&rsquo;s SuperHyperBipartite it&rsquo;s only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming two separate sets, has no SuperHyperEdge in common; it&rsquo;s SuperHyperMultiPartite it&rsquo;s only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no SuperHyperEdge in common; it&rsquo;s SuperHyperWheel if it&rsquo;s only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex. The SuperHyperModel proposes the specific designs and the specific architectures. The SuperHyperModel is officially called &ldquo;SuperHyperGraph&rdquo; and &ldquo;Neutrosophic SuperHyperGraph&rdquo;. In this SuperHyperModel, The &ldquo;specific&rdquo; cells and &ldquo;specific group&rdquo; of cells are SuperHyperModeled as &ldquo;SuperHyperVertices&rdquo; and the common and intended properties between &ldquo;specific&rdquo; cells and &ldquo;specific group&rdquo; of cells are SuperHyperModeled as &ldquo;SuperHyperEdges&rdquo;. Sometimes, it&rsquo;s useful to have some degrees of determinacy, indeterminacy, and neutrality to have more precise SuperHyperModel which in this case the SuperHyperModel is called &ldquo;neutrosophic&rdquo;. In the future research, the foundation will be based on the &ldquo;Cancer&rsquo;s Recognitions&rdquo; and the results and the definitions will be introduced in redeemed ways. The recognition of the cancer in the long-term function. The specific region has been assigned by the model [it&rsquo;s called SuperHyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn&rsquo;t be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it&rsquo;s said to be neutrosophic SuperHyperGraph] to have convenient perception on what&rsquo;s happened and what&rsquo;s done. There are some specific models, which are well-known and they&rsquo;ve got the names, and some SuperHyperGeneral SuperHyperModels. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the longest SuperHyperAlliances or the strongest SuperHyperAlliances in those neutrosophic SuperHyperModels. For the longest SuperHyperAlliances, called SuperHyperAlliances, and the strongest SuperHyperCycle, called neutrosophic SuperHyperAlliances, some general results are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths have only two SuperHyperEdges but it&rsquo;s not enough since it&rsquo;s essential to have at least three SuperHyperEdges to form any style of a SuperHyperCycle. There isn&rsquo;t any formation of any SuperHyperCycle but literarily, it&rsquo;s the deformation of any SuperHyperCycle. It, literarily, deforms and it doesn&rsquo;t form. A basic familiarity with SuperHyperGraph theory and neutrosophic SuperHyperGraph theory are proposed.