A graph with respect to idempotents of a Ring-II

Abstract

Let [Formula: see text] be a ring with unity. The idempotent graph [Formula: see text] of a ring [Formula: see text] is an undirected simple graph whose vertex set is [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is an idempotent element of [Formula: see text]. Razaghi et al. (A graph with respect to idempotents of a ring. J. Algebra Appl., 20(6):2150105, 8, 2021) studied basic properties of [Formula: see text] such as connectedness, diameter and girth. In this article, first we correct a structural result obtained by Razaghi et al. and determine the precise structure of the idempotent graph of local rings. Further, we obtain a necessary and sufficient condition on the ring [Formula: see text] such that [Formula: see text] is planar. We prove that [Formula: see text] is an outerplanar graph if and only if [Formula: see text] is a local ring. Moreover, we classify all the finite commutative rings [Formula: see text] such that [Formula: see text] is claw-free, cograph, split graph and threshold graph, respectively. We conclude that for a finite non-local commutative ring, the latter two graph classes of [Formula: see text] are equivalent if and only if [Formula: see text] is a Boolean ring.