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

摘要

In this paper, we study some graph-theoretic properties about the zero-divisor graph Gamma(Q) of a finite quasi-ordered set Q with a least element 0 and its line graph L(Gamma(Q)). 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 Gamma(Q) by the minimal prime ideals of Q. Resides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete k-partite graphs. We also investigate the planarity of Gamma(Q). Finally, we obtain the characterization for the line graph L(Gamma(Q)) in terms of its diameter, girth and planarity.