On generalized zero divisor graph of a poset

In this paper, we introduce the generalized ideal based zero divisor graph of a poset PP, denoted by GI(P)̂. A representation theorem is obtained for generalized zero divisor graphs. It is proved that a graph is complete rr-partite with r⩾2r⩾2 if and only if it is a generalized zero divisor graph of a poset. As a consequence of this result, we prove a form of a Beck’s Conjecture for generalized zero divisor graphs of a poset. Further, it is proved that a generalized zero divisor graph G{0}(P)̂ of a section semi-complemented poset PP with respect to the ideal (0](0] is a complete graph.