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Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

Riccardo Pengo, Daniel Vallières

TL;DR

The paper develops a global framework to understand how spanning-tree counts evolve in $ abla$-towers of graphs by associating to a base graph a reciprocal Ihara polynomial and its Pierce-Lehmer factor $J_oldsymbol{ ho}$. It shows that $ au(X_n)$ can be expressed through Pierce-Lehmer sequences $oldsymbol{\Delta}_n(J_oldsymbol{ ho})$, with Archimedean growth governed by the Mahler measure $M_ ext{∞}(I_oldsymbol{ ho})$ and $p$-adic growth governed by $M_p(J_oldsymbol{ ho})$, yielding precise asymptotics and $p$-adic valuations. The main results provide explicit formulas for the number of spanning trees in finite abelian covers and connect these counts to Artin-Ihara $L$-functions and the Ihara polynomial, including exact expressions, illustrative examples, and a Fibonacci-tower specialization that recovers Lengyel’s $p$-adic valuation formulas. The framework generalizes prior circulant and Petersen-graph results and offers a path to uniform Iwasawa-type behavior in graph towers via determinant- and resultant-based invariants. This has potential implications for understanding combinatorial growth, zeta functions of graphs, and $p$-adic properties in combinatorial towers.

Abstract

Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and $I$-graphs (including the generalized Petersen graphs). We also express the $p$-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the $p$-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb{Z}$-cover, our result gives us back Lengyel's calculation of the $p$-adic valuations of Fibonacci numbers.

Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

TL;DR

The paper develops a global framework to understand how spanning-tree counts evolve in -towers of graphs by associating to a base graph a reciprocal Ihara polynomial and its Pierce-Lehmer factor . It shows that can be expressed through Pierce-Lehmer sequences , with Archimedean growth governed by the Mahler measure and -adic growth governed by , yielding precise asymptotics and -adic valuations. The main results provide explicit formulas for the number of spanning trees in finite abelian covers and connect these counts to Artin-Ihara -functions and the Ihara polynomial, including exact expressions, illustrative examples, and a Fibonacci-tower specialization that recovers Lengyel’s -adic valuation formulas. The framework generalizes prior circulant and Petersen-graph results and offers a path to uniform Iwasawa-type behavior in graph towers via determinant- and resultant-based invariants. This has potential implications for understanding combinatorial growth, zeta functions of graphs, and -adic properties in combinatorial towers.

Abstract

Using the special value at of Artin-Ihara -functions, we associate to every -cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and -graphs (including the generalized Petersen graphs). We also express the -adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the -adic Mahler measure of the Ihara polynomial. When applied to a particular -cover, our result gives us back Lengyel's calculation of the -adic valuations of Fibonacci numbers.
Paper Structure (16 sections, 15 theorems, 159 equations)

This paper contains 16 sections, 15 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Historical remarks
  3. Main results
  4. Notations and conventions
  5. Mahler measures and Pierce-Lehmer sequences
  6. Resultant
  7. Mahler measure
  8. Pierce-Lehmer sequences
  9. Graph theory
  10. Galois covers of locally finite graphs
  11. The number of spanning trees in finite abelian covers of finite graphs
  12. Exact formulas for the number of spanning trees
  13. An explicit example
  14. Asymptotics for the number of spanning trees
  15. Exact formulas for the p-adic valuation of the number of spanning trees
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a finite connected graph whose Euler characteristic $\chi(X)$ does not vanish, and $\alpha \colon \mathbf{E}_X \to \mathbb{Z}$ be a voltage assignment such that for every $n \geq 1$ the finite graph is connected (which can be checked using connectedness). Let $\mathcal{I}_\alpha \in \mathbb{Z}[t^{\pm 1}]$ be the Ihara polynomial associated to $\alpha$, and set where $b := -\mathrm{ord

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • ...and 35 more