The Zero Divisor Graph of Partially Ordered Set with Respect to a Semi-Ideal

Abstract

In this paper, we investigate the zero divisor graph \(G_I(P)\) of a poset \(P\) with respect to a semi-ideal \(I\). We show that the girth of \(G_I(P)\) is \(3\), \(4\), or \(\infty\). In addition, it is shown that the diameter of such a graph is either \(1\), \(2\), or \(3\). Moreover, we investigate the properties of a cut vertex in \(G_I(P)\) and study the relation between semi-ideal \(I\) and the graph \(G_I(P)\), as established in (Theorem 3.9).