Calculating Different Topological Indices of Von Neumann Regular Graph of Z_(p^α )

Abstract

By the Von Neumann regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x,y if and only if x+y is a von Neumann regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann regular elements by V nr(R). Topological indices are the numbers that is devoted to graphs and show some of their properties. In this paper, first we obtain the degree of vertices for a ring R and the number of edges in different special cases for the ring Z_(p^α ) (p is a prime number) and then we compute Zagreb indices of type one, two and three, Randic, Wiener, Hyper Wiener and reverse Wiener of Von Neumann graph.