On abelian $\ell$-towers of multigraphs

TL;DR

This paper investigates how the $ ext{l}$-adic valuation of the number of spanning trees evolves in regular abelian $ ext{l}$-towers of multigraphs, revealing parallels with class number behavior in number field extensions.

Contribution

It establishes a connection between the $ ext{l}$-adic valuation of spanning trees in multigraph towers and class number phenomena in algebraic number theory.

Findings

$ ext{l}$-adic valuation of spanning trees shows predictable patterns in abelian $ ext{l}$-towers.

Behavior of spanning trees parallels class number growth in $ ext{Z}_{ ext{l}}$-extensions.

Results apply to infinite families of bouquets, illustrating broad applicability.

Abstract

We study how the $\ell$-adic valuation of the number of spanning trees varies in regular abelian $\ell$-towers of multigraphs. We show that for an infinite family of regular abelian $\ell$-towers of bouquets, the behavior of the $\ell$-adic valuation of the number of spanning trees behave similarly to the $\ell$-adic valuation of the class numbers in $\mathbb{Z}_{\ell}$-extensions of number fields.