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 a 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 provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertices and $w(x)\ge 2$ if ${deg}_D(x)\ge 2$ for all $x\in V(D)$. Finally, if $D$ is a weighted rooted tree with the degree of root is $1$ and $w(x)\ge 2$ when ${deg}_D(x)\ge 2$ for all $x\in V(D)$, we show that ${I(D)}^{(s)}={I(D)}^s$ for all $s\ge 2$.

让$D$是一个加权导向图并且$我（D）$是其边缘理想。我们提供一种方法来找到所有最小生成器${我}_{\subseteq C}$， 在哪里$C$是最大强顶点覆盖$D$和${我}_{\subseteq C}$是与包含在中的强顶点覆盖相关的不可约理想的交集$C$。如果$D^{\prime }$是一个诱导有向图$D$，在一定条件下的强顶点覆盖$D^{\prime }$和$D$，我们证明${我（D^{\prime }）}^{（s）}\ne {我（D^{\prime }）}^s$对于一些$s\ge 2$暗示${我（D）}^{（s）}\ne {我（D）}^s$。我们为具有公共汇点的两个自然定向路径的并集的边理想的普通幂和符号幂相等提供了充分必要条件。我们描述了所有最大强顶点覆盖$D$使得至多一条边朝向其每个顶点并且$w（X）\ge 2$如果${度}_D（X）\ge 2$对全部$X\epsilon V（D）$。最后，如果$D$是一棵带权有根树，其根度为$1$和$w（X）\ge 2$什么时候${度}_D（X）\ge 2$对全部$X\epsilon V（D）$，我们证明${我（D）}^{（s）}={我（D）}^s$对全部$s\ge 2$。