Some properties about the zero-divisor graphs of quasi-ordered sets

Abstract

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.