The weakly zero-divisor graph of a commutative ring

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph W Γ(R) with vertex set Z(R) * , and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0.It follows that W Γ(R) contains the zero-divisor graph Γ(R) as a subgraph.In this paper, the connectedness, diameter, and girth of W Γ(R) are investigated.Moreover, we determine all rings whose weakly zero-divisor graphs are star.We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical.Finally, the chromatic number of W Γ(R) is studied.