Generalized zero-divisor graph of ∗-rings

A ∗-ring [Formula: see text] is a ring with an involution ∗. Let [Formula: see text] denote the set of all nonzero zero-divisors of [Formula: see text]. We associate a simple (undirected) graph [Formula: see text] with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] or [Formula: see text], for some positive integer [Formula: see text]. We find the diameter and girth of [Formula: see text]. The characterizations are obtained for ∗-rings having [Formula: see text] a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring [Formula: see text], there is an involution on [Formula: see text] such that [Formula: see text] is disconnected if and only if [Formula: see text] is an integral domain.