Resumen de The weakly zero-divisor graph of a commutative ring

M.J. Nikmehr, A. Azadi, R. Nikandish

• 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 WΓ(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 WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of WΓ(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 WΓ(R) is studied.