On $\mathbb{Z}_{\ell }^{d}$-towers of graphs

DuBose, Sage; Vallières, Daniel

doi:10.5802/alco.304

On

ℤ_{ℓ}^{d}

-towers of graphs

DuBose, Sage ¹; Vallières, Daniel ¹

¹ California State University Mathematics and Statistics Department Chico CA 95929 USA

Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1331-1346.

Abstract

Let $ℓ$ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when $n$ is sufficiently large, the $ℓ$ -adic valuation of the number of spanning trees at the $n$ th layer of a $ℤ_{ℓ}^{d}$ -tower of graphs is given by a polynomial in $ℓ^{n}$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.

Received: 2022-07-04
Revised: 2023-01-18
Accepted: 2023-02-15
Published online: 2023-11-07

DOI: 10.5802/alco.304

Classification: 05C25, 11R18, 11R23, 11Z05
Keywords: Ihara zeta functions, Iwasawa theory, spanning trees

Author's affiliations:

DuBose, Sage ¹; Vallières, Daniel ¹

¹ California State University Mathematics and Statistics Department Chico CA 95929 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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DuBose, Sage; Vallières, Daniel. On $\mathbb{Z}_{\ell }^{d}$-towers of graphs. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1331-1346. doi : 10.5802/alco.304. https://alco.centre-mersenne.org/articles/10.5802/alco.304/

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