Nil clean graphs over Z p rings

Abstract

Let R be a commutative ring with identity and N.C(R) be the set of all nil clean elements of R. In this paper, we examine the nil clean graph of R, denoted by GN(R), that vertices of GN(R) are all nonzero elements of R and two distinct vertices x and y are adjacent if and only xy > N.C(R). We investigate the properties of the nil clean graph GN(R) where R is a direct product of Zpi ’s for some prime numbers p1, p2,⋯, pt. We obtain some graph theoretic properties of the nil clean graph like diameter, girth, clique number, chromatic number.