Generalized zero-divisor graph of *-rings

Let R be a ring with involution * and Z^* (R) denotes the set of all non-zero zero-divisors of R. We associate a simple (undirected) graph ' (R) with vertex set Z^* (R) and two distinct vertices x and y are adjacent in ' (R) if and only if xⁿy^*=0 or yⁿx^*=0, for some positive integer n. We find the diameter and girth of ' (R). The characterizations are obtained for *-rings having ' (R) a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring R, there is an involution on R R such that ' (R R) is disconnected if and only if R is an integral domain.