0054 | Failed Clique Number in Neutrosophic Graphs

New setting is introduced to study neutrosophic failed-clique number and failed clique neutrosophic-number arising being out of neighborhood of vertices. Being out of neighbor is a key term to have these notions. Not having all possible edges amid vertices in a set is a key type of approach to have these notions namely neutrosophic failed-clique number and failed clique neutrosophic-number. Two numbers are obtained but now both settings leads to approach is on demand which is finding smallest set which doesn’t have any vertex which is neighbor. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then failed clique number CF(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is minimum cardinality of a set S of vertices such that there are two vertices in S aren’t endpoints for an edge, simultaneously; failed clique neutrosophic-number CnF(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is minimum neutrosophic cardinality of a set S of vertices such that there are two vertices in S aren’t endpoints for an edge, simultaneously. As concluding results, there are some statements, remarks, examples and clarifications about some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs and complete-t-partite-neutrosophic graphs. The clarifications are also presented in both sections “Setting of Neutrosophic Failed-Clique Number,” and “Setting of Failed Clique Neutrosophic-Number,” for introduced results and used classes. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number to compare with other vertices. Summation of three values of vertex makes one number and applying it to a comparison. This approach facilitates identifying vertices which form neutrosophic failed-clique number and failed clique neutrosophic-number arising being out of neighborhood of vertices. In path-neutrosophic graphs, two vertices which aren’t neighbors, form minimal set but with slightly differences, in cycle-neutrosophic graphs, two vertices which aren’t neighbors form minimal set. Other classes have same approaches. In complete-neutrosophic graphs, an empty set leads us to neutrosophic failed-clique number and failed clique neutrosophic-number. In star-neutrosophic graphs, a set of vertices containing only two vertices which aren’t neighbors, makes minimal set. In complete-bipartite-neutrosophic graphs, a set of vertices including two vertices from same part makes intended set but with slightly differences, in complete-t-partite-neutrosophic graphs, a set of two vertices from same part makes intended set. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set to extend this set to set of all vertices has key role to have these notions in the form of neutrosophic failed-clique number and failed clique neutrosophic-number arising being out of neighborhood of vertices. The cardinality of a set has eligibility to neutrosophic failed-clique number but the neutrosophic cardinality of a set has eligibility to call failed clique neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, two vertices have no connection amid each other, opens the way to do some approaches. A vertex could affect on other vertex but there’s no usage of edges. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.