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 Gamma(R) with vertex set Z(R)*, and two distinct vertices x and y are adjacent if and only if there exist r is an element of ann(x) and s is an element of ann(y) such that rs = 0. It follows that W Gamma(R) contains the zero-divisor graph Gamma(R) as a subgraph. In this paper, the connectedness, diameter, and girth of W Gamma(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 Gamma(R) is studied.