Enumeration of Graph Coverings and Their Generalization

图覆盖的枚举及其泛化

基本信息

批准号：
11640145
负责人：
SATO Iwao
金额：
$ 1.09万
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：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640145/
关键词：
graph covering enumeration 皮覆グラフ

项目摘要

We consider four objects in enumeration of graph coverings and their generalization : enumeration of regular coverings ; enumeration of g-cyclic A-covers ; lifts of automorphisms of symmetric digraphs; zeta functions of regular coverings.The general problem of counting the ismorphism classes of regular n-fold coverings of a graph G with respect to a group Γ of automorphisms of G is still unsolved except in the case that n is prime. The enumeration of Γ-isomorphism classes of regular p^n-fold coverings of G is a natural problem. A regular p^2-fold covering of G is either a Z_p×Z_p-covering or a Z_<p2>-covering of G. We enumerate the Γ-isomorphism classes of Z_p×Z_p-coverings of G.Furthermore, we show that it is possible to count the Γ-isomorphism classes of Z_p^n-coverings of G for any prime p(>2) and any 3≦n≦p.Next, for a connected symmetric digraph D, a finite group A and g∈A, we consider a g-cyclic A-cover of D as a generalization of a regular covering of a graph. We enumerate the Γ-isomorphism classes of g-cyclic Z_p×Z_p-covers and g-cyclic Z_p^3-covers of D. In the case that A is an abelian group, we present a characterization for two g-cyclic A-covers of D to be ismorphic with respect to a group Γ of automorphisms of D. Thus, we enumerate the I-isomorphism classes of g-cyclic Z_2^n-covers and g-cyclic Z_<2n>-covers of D.For a group Γ of automorphisms of a symmetric digraph D, we present a necessary and sufficient condition for Γ to have a lift with respect to a cyclic A-cover of D, and characterize the lift of Γ to be a split extension of A by Γ.As an application of a decomposition formula for the characteristic polynomial of a regular covering of a graph G, we obtain a factorization of the zeta function of a regular covering of G. Furthermore, we factorize the zeta function of a g-cyclic A-cover of a symmetric digraph.

我们考虑了列举图表及其概括的四个对象：定期覆盖物的列举； G-cycle A覆盖的枚举；对称挖掘的自动形态的升力；定期覆盖的ZETA功能。计数Graph G的常规n倍覆盖物相对于G的γ的常规n折类别的总体问题仍然未解决，除非N是素数。 g的常规p^n倍覆盖物的γ-异态类别是z_p×z_p覆盖或z__ <p2> - G.我们列举了z_p×z_p z_p z_p z_p z_p z_p z_p z_p z_p z_-iSomorthism com g。对于任何prime p（> 2）和任何3 nn≦p.next的g，对于连接的对称挖掘d，有限的组A和g∈A，我们将d的g-cycle a-cover视为图形常规覆盖率的概括。我们列举了G周期ZP×ZP×ZP覆盖和G-Cycle Z_p^3封面的γ-异态类别。 Z_2^n-covers和g-cycle z_ <2n> - 对于对称挖掘的自动形态的γγ的covers，我们提出了γ的必要条件，使γ具有d lift aiff a comclic a-cover d of D的demplationd fip a comclation d fips a lift的升力，并表征γ的特征范围，以使γ的特征范围扩展，以γ.as的特征范围，以γ.as的特征范围，以γ.as的特征范围。在图G中，我们获得了G的常规覆盖的Zeta函数的分解。此外，我们将对称挖掘的G-Cycle A覆盖的Zeta函数分解。