On the Structure of Nil Graph of a Commutative Ring

Abstract

Let R be a commutative ring and N(R) be the set of all non-zero nil elements of index two. The Nil Graph of R, denoted by Γ_N(R), is an undirected graph with the vertex set Z_N(R)^*={x ϵ R*| xy ϵ N(R) for some y in R*=R-{0}} and any two vertices of Z_N(R)^* are adjacent if and only if xy ϵ N(R). In this paper we determine the chromatic number of the nil graph Γ_N (Z_(p^\alpha q)) . Also we study the diameter and girth of Γ_N (Z_(p^\alpha q)).