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Spanning trees and their relations in Galois covers

Kosuke Mizuno

TL;DR

The paper generalizes spanning-tree relations in Galois covers from the $(\mathbb{Z}/2\mathbb{Z})^{m}$ case to arbitrary finite groups by proving two Möbius-inversion formulas that express the spanning-tree count of the top and intermediate graphs via intermediate covers. Using the Ihara zeta function and Artin–Ihara $L$-functions, the authors derive graph-theoretic analogues of Kuroda's formula and Brauer–Kuroda relations, establishing a broad framework that links intermediate-graph complexities to the global complexity $\kappa(Y)$. A key insight is the role of Möbius functions on the posets $\mathscr{H}_G$ and $\mathscr{C}_G$ in weighting these products, which recovers the known HKSV result in the $\mathbb{Z}/2\mathbb{Z}$-power case and clarifies when such formulas do not exist, notably for cyclic G. The results deepen the connection between graph theory and algebraic number theory, provide practical means to compute $\kappa(Y)$ from intermediate-data when $G$ is noncyclic, and reveal intrinsic limitations for monomial-type spanning-tree relations in the cyclic setting.

Abstract

This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^m$-covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin--Ihara $L$-function, we prove two formulas which are graph-theoretic analogues of Kuroda's formula and the Brauer--Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic.

Spanning trees and their relations in Galois covers

TL;DR

The paper generalizes spanning-tree relations in Galois covers from the case to arbitrary finite groups by proving two Möbius-inversion formulas that express the spanning-tree count of the top and intermediate graphs via intermediate covers. Using the Ihara zeta function and Artin–Ihara -functions, the authors derive graph-theoretic analogues of Kuroda's formula and Brauer–Kuroda relations, establishing a broad framework that links intermediate-graph complexities to the global complexity . A key insight is the role of Möbius functions on the posets and in weighting these products, which recovers the known HKSV result in the -power case and clarifies when such formulas do not exist, notably for cyclic G. The results deepen the connection between graph theory and algebraic number theory, provide practical means to compute from intermediate-data when is noncyclic, and reveal intrinsic limitations for monomial-type spanning-tree relations in the cyclic setting.

Abstract

This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for -covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin--Ihara -function, we prove two formulas which are graph-theoretic analogues of Kuroda's formula and the Brauer--Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic.

Paper Structure

This paper contains 12 sections, 18 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Graph Theory
  3. Definition of graphs
  4. Galois theory for graphs
  5. Zeta Functions for Graphs
  6. Ihara zeta functions
  7. Artin--Ihara $L$-functions
  8. Spanning tree formulas for Galois covers
  9. The generalization of Kuroda's formula
  10. The Brauer--Kuroda relations for graphs
  11. Non-existence of a spanning tree formula for cyclic Galois groups
  12. The Möbius Functions of Finite Posets

Key Result

Theorem 1.1

Let $G$ be a finite group, and let $\mathscr{H}_{G}$ be the partially ordered set defined above. Let $\mu\colon \ul{\mathscr{H}_{G}}\times\ul{\mathscr{H}_{G}}\to \mathbb{Z}$ be the Möbius function of $\ul{\mathscr{H}_{G}}$ (cf. Appendix mobi fu). Then, for any Galois cover $Y/X$ with $\mathop{\mathr where $X_H$ denotes the intermediate graph corresponding to $H$.

Figures (3)

  • Figure 4.1: The derived graph $X(\alpha)$ (left) and a bouquet graph $X$ with two loops (right).
  • Figure 4.2: The Hasse diagram of $\ul{\mathscr{H}_{G}}$ (left) and the intermediate graphs of $Y/X$ (right).
  • Figure 4.3: The Hasse diagram of $\overline{\mathscr{C}_{G}}$ (left) and the intermediate graphs of $Y/X$ (right).

Theorems & Definitions (44)

  • Theorem 1.1: $=$ \ref{['main1']}
  • Theorem 1.2: $=$ \ref{['main2']}
  • Theorem 1.3: $=$ \ref{['main3']}
  • Definition 2.1: Ser77, Sunada
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 3.1: cf. Terras
  • Proposition 3.2: cf. pengo
  • ...and 34 more