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The graph zeta functions with respect to the group matrix of a finite group

Tsuyoshi Miezaki, Iwao Sato

TL;DR

This work studies zeta-type graph invariants tied to the group matrix $M(\Gamma)$ of a finite abelian group $\Gamma$. By deriving explicit determinant expressions for the edge zeta function and the second weighted zeta function with respect to $M(\Gamma)$, it links graph zeta theory to the representation-theoretic structure of $\Gamma$. A central contribution is a new, conceptually transparent proof of Dedekind's theorem for the group determinant, achieved via decomposition formulas for group-covering digraphs and, in the abelian case, character sums. The paper further connects these zeta-inspired methods to the weighted complexity of the complete graph when arc weights come from the group matrix, providing concrete formulas and recovering classical results in the unweighted limit.

Abstract

In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $Γ$. Furthermore, we give another proof of Dedekind Theorem for the group determinant of $Γ$ by the decomposition formula for a matrix of a group covering of a digraph. Finally, we treat the weighted complexity of the complete graph with entries of the group matrix of $Γ$ as arc weights.

The graph zeta functions with respect to the group matrix of a finite group

TL;DR

This work studies zeta-type graph invariants tied to the group matrix of a finite abelian group . By deriving explicit determinant expressions for the edge zeta function and the second weighted zeta function with respect to , it links graph zeta theory to the representation-theoretic structure of . A central contribution is a new, conceptually transparent proof of Dedekind's theorem for the group determinant, achieved via decomposition formulas for group-covering digraphs and, in the abelian case, character sums. The paper further connects these zeta-inspired methods to the weighted complexity of the complete graph when arc weights come from the group matrix, providing concrete formulas and recovering classical results in the unweighted limit.

Abstract

In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group . Furthermore, we give another proof of Dedekind Theorem for the group determinant of by the decomposition formula for a matrix of a group covering of a digraph. Finally, we treat the weighted complexity of the complete graph with entries of the group matrix of as arc weights.

Paper Structure

This paper contains 8 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zeta functions of a graph
  4. The group matrix
  5. The edge zeta function with respect to the group matrix
  6. The second weighted zeta function with respect to the group matrix
  7. Another proof for Dedekind Theorem
  8. The weighted complexity of the complete graph with weights with respect to a group matrix

Key Result

Theorem 1

Let $G$ be a connected simple graph with $n$ vertices and $m$ edges. Then the reciprocal of the Ihara zeta function of $G$ is given by where ${\bf A} (G)$ is the adjacency matrix of $G$, and ${\bf D} =( d_{ij} )$ is the diagonal matrix with $d_{ii} = \deg v_i$ where $V(G)= \{ v_1 , \ldots , v_n \}$.

Theorems & Definitions (15)

  • Theorem 1: Hashimoto; Bass
  • Theorem 2: Stark and Terras
  • Theorem 3: Watanabe and Fukumizu
  • Theorem 4: Sato
  • Theorem 5: Dedekind
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Corollary 1
  • Corollary 2
  • ...and 5 more