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Zeta functions of graphs and coverings

图和覆盖层的 Zeta 函数

基本信息

批准号：
15540147
负责人：
SATO Iwao
金额：
$ 1.15万
依托单位：
Oyama National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540147/
关键词：
graph covering zeta function 被膜グラフ

项目摘要

We express an L-function of a regular covering of a graph G by using that of G. Moreover, we treat zeta functions and L-functions of a semiregular bipartite graph, its line graph, its middle graph and their regular coverings, and present an analogue of the Selberg trace formula for an L-function of a semiregular bipartite graph.We consider weighted zeta functions and L-functions of digraphs, and give their determinant expressions. Moreover, we present determinant expressions for weighted zeta functions and weighted L-functions of graphs, and express the weighted zeta function of a regular covering of a graph as a product of its weighted L-functions. By using a similar method to the above one, we present determinant expressions for the weighted complexities of a graph and its (regular or irregular) covering. Furthermore, we present a new decomposition formula for the weighted zeta function of a (regular or irregular) covering of a graph, and study the structure of a balanced covering of a unbalanced graph as an application.Finally, we consider the Bartholdi zeta functions and Bartholdi L-functions of a graph, a digraph and their covering, and present their determinant expressions and decomposition formulas. Furthermore, we present decomposition formulas for the Bartholdi zeta function of some branched covering of a graph, and the weighted Bartholdi zeta function of a graph.

我们用 G 的函数来表达图 G 的正则覆盖的 L 函数。此外，我们处理半正则二分图、其线图、中间图及其正则覆盖的 zeta 函数和 L 函数，并提出半正则二分图的 L 函数的 Selberg 迹公式的类似物。我们考虑加权 zeta 函数和有向图的 L 函数，并给出它们行列式表达式。此外，我们提出了图的加权 zeta 函数和加权 L 函数的行列式，并将图的正则覆盖的加权 zeta 函数表示为其加权 L 函数的乘积。通过使用与上述方法类似的方法，我们提出了图及其（规则或不规则）覆盖的加权复杂度的行列式。此外，我们提出了图的（规则或不规则）覆盖的加权zeta函数的新分解公式，并研究了不平衡图的平衡覆盖的结构作为应用。最后，我们考虑了图、有向图及其覆盖的Bartholdi zeta函数和Bartholdi L函数，并给出了它们的行列式和分解公式。此外，我们提出了图的某些分支覆盖的 Bartholdi zeta 函数的分解公式，以及图的加权 Bartholdi zeta 函数。