Finding Hamiltonian Neutrosophic Cycles in Classes of Neutrosophic Graphs

New setting is introduced to study hamiltonian neutrosophic cycle and n-hamiltonian neutrosophic cycle arising from finding and counting longest neutrosophic cycles containing all vertices once in strong neutrosophic graphs based on neutrosophic cycles and in neutrosophic graphs based on crisp cycles. Forming neutrosophic cycles from a sequence of consecutive vertices is key type of approach to have these notions namely hamiltonian neutrosophic cycle and n-hamiltonian neutrosophic cycle arising from finding and counting longest neutrosophic cycles containing all vertices once in strong neutrosophic graphs based on neutrosophic cycles and in neutrosophic graphs based on crisp cycles. One number and one sequence are obtained but now both settings leads to approach is on demand which is counting minimum cardinality and a sequence in the terms of vertices, which have edges which form neutrosophic cycle and crisp cycles concerning finding and counting longest neutrosophic cycles containing all vertices once. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then hamiltonian neutrosophic cycle M(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is a sequence of consecutive vertices x1,x2,··· ,xO(NTG),x1 which is neutrosophic cycle; n-hamiltonian neutrosophic cycle N(HNC) for a neutrosophic graph NTG : (V,E,σ,μ) is the number of sequences of consecutive vertices x1,x2,··· ,xO(NTG),x1 which are neutrosophic cycles. As concluding results, there are some statements, remarks, examples and clarifications about some classes of strong neutrosophic graphs namely (strong-)path-neutrosophic graphs, (strong-)cycle-neutrosophic graphs, complete-neutrosophic graphs,

(strong-)star-neutrosophic graphs, (strong-)complete-bipartite-neutrosophic graphs, (strong-)complete-t-partite-neutrosophic graphs and (strong-)wheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of hamiltonian

neutrosophic cycle,” and “Setting of n-hamiltonian neutrosophic cycle,” 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 hamiltonian neutrosophic cycle and n-hamiltonian neutrosophic cycle arising from finding and counting longest neutrosophic cycles containing all vertices once in strong neutrosophic graphs based on neutrosophic cycles and in neutrosophic graphs based on crisp cycles. In both settings, some classes of well-known (strong) neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The cardinality of a set has eligibility

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to define n-hamiltonian neutrosophic cycle but the sequence has eligibility to define hamiltonian neutrosophic cycle. Some results get more frameworks and perspective about these definitions. The way in that, a sequence of consecutive vertices forming a longest neutrosophic cycles containing all vertices once, opens the way to do some approaches. These notions are applied into strong neutrosophic graphs and neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special strong neutrosophic graphs and 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.