On the ideal-based triple zero-divisor graph of commutative ring

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a proper ideal of a commutative ring [Formula: see text]. The ideal-based triple zero-divisor graph of a commutative ring is a graph, denoted by [Formula: see text], with the vertex set [Formula: see text] and two vertices [Formula: see text] are adjacent if and only if there is a [Formula: see text] such that [Formula: see text]. In this paper, we discuss the connectedness, diameter, girth of [Formula: see text]. We classify all finite commutative rings for which [Formula: see text] is either complete, unicyclic or split graph. Also, we characterize all finite commutative rings for which [Formula: see text] is perfect. Finally, we classify all finite commutative rings for which [Formula: see text] has genus at most one.