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For a commutative ring R, the zero-divisor graph of R is the graph whose vertices are the nonzero zerodivisors of R such that the vertices x and y are adjacent if and only if xy = 0. In this paper, we classify the zero-divisor graphs of Boolean rings, as well as those of Boolean rings that are rationally complete. We also provide a complete list of those rings whose zero-divisor graphs have the property that every vertex is either an end or adjacent to an end.

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In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

An algorithm is presented for constructing the zero-divisor graph of a direct product of integral domains. Moreover, graphs which are realizable as zero-divisor graphs of direct products of integral domains are classified, as well as those of Boolean rings. In particular, graphs which are realizable as zero-divisor graphs of finite reduced commutative rings are classified.