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

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.