Properties of symbolic powers of edge ideals of weighted oriented graphs

Abstract

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $I_{\subseteq C}$, where $C$ is a maximal strong vertex cover of $D$ and $I_{\subseteq C}$ is the intersections of irreducible ideals associated to the strong vertex covers contained in $C$. If $D^{\prime }$ is an induced digraph of $D$, under certain condition on the strong vertex covers of $D^{\prime }$ and $D$, we show that ${I(D^{\prime })}^{(s)}\ne {I(D^{\prime })}^s$ for some $s\ge 2$ implies ${I(D)}^{(s)}\ne {I(D)}^s$. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertex and $w(x)\ge 2$ if ${°}_D(x)\ge 2$ for all $x\in V(D)$. If $D$ is a weighted rooted tree with degree of root is $1$ and $w(x)\ge 2$ when ${°}_D(x)\ge 2$ for all $x\in V(D)$, we show that ${I(D)}^{(s)}={I(D)}^s$ for all $s\ge 2$