Let D be a weighted oriented graph with the underlying graph G when vertices with non-trivial weights are sinks and I(D),I(G) be the edge ideals corresponding to D and G, respectively. We give an explicit description of the symbolic powers of I(D) using the concept of strong vertex covers. We show that the ordinary and symbolic powers of I(D) and I(G) behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of I(D) for certain classes of weighted oriented graphs. When D is a weighted oriented odd cycle, we compute reg(I(D)(s)/I(D)s) and prove regI(D)(s) ≤regI(D)s and show that equality holds when there is only one vertex with non-trivial weight.

中文翻译：

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加权有向图中的符号幂

让D是带有底层图的加权导向图G当具有非平凡权重的顶点是汇点并且一世(D),一世(G)是对应于的边缘理想D和G,分别。我们对符号的力量给出了明确的描述一世(D)使用强顶点覆盖的概念。我们证明了普通的和象征性的力量一世(D)和一世(G)以类似的方式行事。我们提供了符号幂和 Waldschmidt 常数的描述一世(D)对于某些类别的加权有向图。什么时候D是一个加权的奇数循环，我们计算注册(一世(D)(s)/一世(D)s)并证明注册一世(D)(s) ≤注册一世(D)s并证明当只有一个具有非平凡权重的顶点时，等式成立。