Poset graphs and the lattice of graph annihilators

In this paper, the concept of a zero-divisor graph is extended to partially ordered sets with a least element 0. A notion of an annihilator set in a graph is introduced, and it is observed that the annihilator sets in a graph form a complete lattice under inclusion. It is proved that a simple connected graph GG with at least two vertices is realizable as the zero-divisor graph of a partially ordered set if and only if the annihilator sets in GG form a Boolean algebra. The special cases of atomic posets and atomic Boolean algebras are also examined.