On the Structure of Nil Graph of a Commutative Ring

Let R be a commutative ring and N(R) be the set of all nil elements of index two. The nil graph of R denoted by ΓN(R), is an undirected graph with the vertex set ZN(R)* = {x (element of) R*|xy (element of) N(R) for some y in R* = R − {0}}, and any two vertices x and y of ZN(R)* are adjacent if and only if xy (element of) N(R).In this paper we determine the chromatic number of the nil graph ΓN(Zpαq) , where Zpαq is the cyclic group of order p^αq. Also we study the diameter and girth of ΓN(Zpαq).