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 $\mathrm{\infty }$. 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).