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 Γ'(R) with vertex set Z^*(R) and two distinct vertices x and y are adjacent in Γ'(R) if and only if xⁿy^*=0 or yⁿx^*=0, for some positive integer n. We find the diameter and girth of Γ'(R). The characterizations are obtained for $*$-rings having Γ'(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 Γ'(R× R) is disconnected if and only if R is an integral domain.