On the ideal-based zero-divisor graph of commutative rings

Abstract

Let [Formula: see text] be a finite commutative ring with identity, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] denotes the Jacobson radical of [Formula: see text]. The ideal-based zero-divisor graph [Formula: see text] of [Formula: see text] is a graph with vertex set [Formula: see text] for some [Formula: see text] in which distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine the diameter, girth of [Formula: see text]. Specifically, we classify all finite commutative nonlocal rings for which [Formula: see text] is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of [Formula: see text] and characterize all of them.