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)} \neq {I(D^{\prime})}^s $ for some $s \geq 2$ implies $ {I(D)}^{(s)} \neq {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 $w(x) \geq 2$ if $\deg_D(x)\geq 2 $ for all $x \in V(D)$. If $ D $ is a weighted rooted tree with degree of root is $ 1 $ and $ w(x) \geq 2 $ when $ \deg_D(x) \geq 2 $ for all $ x \in V(D) $, we show that $ {I(D)}^{(s)} = {I(D)}^s $ for all $s \geq 2$