有向图的顶点加权zeta函数

给定一个有向图，每个顶点赋予一个权重。对于这个有向图定义了它的顶点加权zeta函数，函数的权重是由上面给定的顶点的权重诱导的有向图的圈的权重。并利用Amitsur恒等式和线性代数的相关知识给出了有向图的顶点加权zeta函数的两个行列式表达式。当有向图是对称有向图时，已有的结果是本文结果的自然推论。最后给出了一个例子并计算出此时顶点加权zeta函数的具体形式。

A vertex weighted zeta function of a digraph

Given a digraph $ G $, a weight was assigned to each vertex. For this digraph, a vertex weighted zeta function was defined, the weight of which was associated with the weights of the cycles in $ G $, induced by the vertex weights given above. Two determinant expressions of the vertex weighted zeta function were provided, using Amitsur identity and some techniques in linear algebra. When the digraph was symmetric, the existing results were natural corollaries of our results. Finally, an example was given and the specific form of the vertex weighted zeta function was calculated.