Nil clean graphs associated with commutative rings

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nilpotent elements of [Formula: see text]. The nil clean graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all nonzero nil clean elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. We prove that [Formula: see text] is connected with [Formula: see text] and [Formula: see text]. We investigate also the interplay between the ring-theoretic properties of [Formula: see text] and the graph-theoretic properties of [Formula: see text]. Moreover, the clique number and the chromatic number of [Formula: see text] for some classes of rings are determined.