A *-ring Formula: see text is a ring with an involution *. Let Formula: see text denotes 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.
Lande et al. (Fri,) studied this question.