0052 | (Failed) 1-independent Number in Neutrosophic Graphs

New setting is introduced to study 1-independent number, 1-independent neutrosophic-number, failed 1-independent number and failed 1-independent neutrosophic-number arising neighborhood of different vertices. 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 1-independent number, 1-independent neutrosophic-number, failed 1-independent number and failed 1-independent neutrosophic-number. Four numbers are obtained but now four settings leads to approach is on demand which is finding biggest set which have all vertices which are(n’t) neighbors. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then 1-independent number I(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum cardinality of a set S of vertices such that every two vertices of S aren’t endpoints for an edge, simultaneously For one time, one vertex is allowed to be endpoint; 1-independent neutrosophic-number In(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum neutrosophic cardinality of a set S of vertices such that every two vertices of S aren’t endpoints for an edge, simultaneously. For one time, one vertex is allowed to be endpoint. Failed 1-independent number I(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum cardinality of a set S of vertices such that every two vertices of S are endpoints for an edge, simultaneously. For one time, one vertex is allowed not to be endpoint; failed 1-independent neutrosophic-number In(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum neutrosophic cardinality of a set S of vertices such that every two vertices of S are endpoints for an edge, simultaneously. For one time, one vertex is allowed not to be endpoint. 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) 1-independent Number,” and “Setting of (Failed) 1-independent 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 1-independent number, 1-independent neutrosophic-number, failed 1-independent number and failed 1-independent neutrosophic-number arising neighborhoods of vertices.

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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 (not) to extend this set to set of all vertices has key role to have these notions in the form of 1-independent number, 1-independent neutrosophic-number, failed 1-independent number and failed 1-independent neutrosophic-number arising neighborhood of vertices. The cardinality of a set has eligibility to neutrosophic (failed) 1-independent number but the neutrosophic cardinality of a set has eligibility to call (failed) 1-independent neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, two vertices (don’t) have connections 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.