A Study on Bipolar Spherical Neutrosophic Cubic Graphs and its Applications

Abstract

In this research work, the concepts of bipolar fuzzy set, bipolar fuzzy graph have been newlineintroduced. The operations on bipolar spherical fuzzy graphs namely, symmetric difference and newlinerejection have been discussed and developed results related to their degrees and total degrees with example. Neutrosophic cubic graph is extended to bipolar environment and combined with spherical fuzzy set to develop a theoretical study, bipolar spherical fuzzy neutrosophic cubic graph. The significant properties of Cartesian product, composition, m-join, n-join, m-union, n-union of a graph have been studied. Application of bipolar spherical fuzzy neutrosophic graph in decision making problem is presented. An algorithm for construction of a minimum spanning tree of bipolar spherical fuzzy graph, bipolar spherical fuzzy neutrosophic cubic graph have been developed. A vertex coloring for spherical fuzzy graph, union, complement of spherical fuzzy graph, strong spherical fuzzy graph, complement of strong spherical fuzzy graph, complete spherical fuzzy graph is defined and developed the chromatic number of spherical fuzzy graph as a crisp number. newlinei) Major objectives : newline and#61472;To define Symmetric difference of bipolar spherical fuzzy graph newline and#61472;To define rejection of bipolar spherical fuzzy graph newline and#61472;To define score function of bipolar spherical neutrosophic cubic set and present bipolar newlinespherical neutrosophic cubic minimum spanning tree algorithm newline and#61472;To define bipolar spherical neutrosophic cubic graph and its algebraic operations such as newlinedegree, order, union, join, and composition. newline and#61472;To define score function of bipolar spherical neutrosophic cubic set and present bipolar newlinespherical neutrosophic cubic minimum spanning tree algorithm. newlineii) Hypothesis: newline and#61472;To provide graphical representation for easy understanding newline and#61472;To provide examples for clear understanding of defined concepts