The non-$\ell$-part of the number of spanning trees in abelian $\ell$-towers of multigraphs

TL;DR

This paper investigates the $p$-adic valuation of spanning trees in abelian $ ext{l}$-towers of multigraphs, drawing analogies with class group growth in cyclotomic extensions, and shows that the number of prime divisors can be unbounded.

Contribution

It introduces a novel analogy between spanning tree counts in graph towers and class group growth in number theory, extending classical results to a graph-theoretic setting.

Findings

$p$-adic valuation of spanning trees studied in abelian $ ext{l}$-towers.

Number of prime divisors of spanning trees can be unbounded.

Analogies with classical number theory results established.

Abstract

Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth of the $p$-part of the class group in a cyclotomic $\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.