On generalized zero divisor graph of a poset

In this paper, we introduce the generalized ideal based zero divisor graph of a poset P, denoted by <(G(I)(P))over cap>. A representation theorem is obtained for generalized zero divisor graphs. It is proved that a graph is complete r-partite with r >= 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))over cap> of a section semi-complemented poset P with respect to the ideal (0] is a complete graph. (C) 2013 Elsevier By. All rights reserved.