Nil clean graphs associated with commutative rings

Abstract

Let R be a commutative ring with identity and Nil(R) be the set of all nilpotent elements of R. The nil clean graph of R, denoted by N.G(R), is a graph whose vertices are all nonzero nil clean elements of R and two distinct vertices x and y are adjacent if and only if xy ∈ Nil(R) or x−y ∈ Nil(R). In this paper, we focus on N.G2(R), the subgraph of N.G(R) induced by the set N.G2(R) = {(e, n) : 0, 1 = e ∈ Id(R) and n ∈ Nil(R)}. It is observed that N.G2(R) has a crucial role in N.G(R). We prove that N.G2(R) is connected with diam(N.G2(R)) ≤ 3 and gr(N.G2(R)) ∈ {3,∞}. We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of N.G2(R). Moreover, the clique number and the chromatic number of N.G2(R) for some classes of rings are determined.