Relations Between Vertex$–$Edge Degree Based Topological Indices and $M_{ve}$-Polynomial of $r-$Regular Simple Graph

Abstract

One of the more exciting polynomials among the newly presented graph algebraic polynomials is the $M-$Polynomial, which is a standard method for calculating degree$-$based topological indices. In this paper, we define the $M_{ve}-$polynomials based on vertex$–$edge degree and derive various vertex$–$edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex$–$edge degree topological indices. Also, we discuss the general $M_{ve}-$polynomial of $r-$regular simple graph. Finally, we computed the $M_{ve}-$polynomial of the $2-$ary tree graph.