Neutrosophic Collapsed Numbers in the Viewpoint of Neutrosophic Graphs

New setting is introduced to study collapsed number and neutrosophic collapsed number arising from smallest neighborhood and the minimum number of neighbors in neutrosophic graphs assigned to neutrosophic graphs. Consider a vertex. Minimum number of neighbors based on that vertex to compare to its neighbors, is a number which is representative based on that vertex. Minimum neutrosophic number of vertices amid neutrosophic cardinality of all sets of vertices is called neutrosophic collapsed number. Forming sets from special vertices to figure out different types of number of vertices having smallest number of neighbors from two vertices are neighbors in the terms of minimum number of vertices to get minimum number to assign to neutrosophic graphs is key type of approach to have these notions namely collapsed number and neutrosophic collapsed number arising from smallest neighborhood and the minimum number of neighbors in neutrosophic graphs assigned to neutrosophic graphs. Two numbers and one set are assigned to a neutrosophic graph, are obtained but now both settings lead to approach is on demand which is to compute and to find representatives of vertices having smallest number of neighbors from two vertices are neighbors forming different types of sets of vertices in the terms of minimum number and minimum neutrosophic number forming it to get minimum number to assign to a neutrosophic graph. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then a set of vertices S is called collapsed set if for every vertex y outside, there’s at least one vertex x inside such that they’re endpoints xy ∈ E and the number of neighbors of x is less than [equal to] the number of neighbors of y. The minimum cardinality between all collapsed sets is called collapsed number and it’s denoted by P(NTG); a set of vertices S is called collapsed set if for every vertex y outside, there’s at least one vertex x inside such that they’re endpoints xy ∈ E and the number of neighbors of x is less than [equal to] the number of neighbors of y. The minimum neutrosophic cardinality 􏰀x∈S 􏰀3i=1 σi(x) between all collapsed sets is called neutrosophic collapsed number and it’s denoted by Pn(NTG). 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, complete-t-partite-neutrosophic graphs and wheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of collapsed number,” and “Setting of neutrosophic collapsed number,” for introduced results and used classes. This approach facilitates identifying vertices which form collapsed number and neutrosophic collapsed number arising from smallest neighborhood and the minimum number of neighbors in neutrosophic graphs assigned to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The cardinality of set of vertices and neutrosophic cardinality of set of vertices have eligibility to define collapsed number and neutrosophic collapsed number but different types of vertices have eligibility to define collapsed sets. Some results get more frameworks and perspective about these definitions. The way in that, different types of vertices having smallest number of neighbors from two vertices are neighbors forming different types of sets in the terms of minimum number of vertices having smallest number of neighbors from two vertices are neighbors and smallest number of vertices having smallest number of neighbors from two vertices are neighbors forming it to get minimum number to assign to neutrosophic graphs or in other words, the way in that, consider a vertex. Minimum number of neighbors based on that vertex to compare to its neighbors, is a number which is representative based on that vertex and minimum neutrosophic number of vertices amid neutrosophic cardinality of sets containing representative numbers is called neutrosophic collapsed number, opens the way to do some approaches. 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. Neutrosophic collapsed notion is applied to different settings and classes of neutrosophic graphs. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.