On the distribution of Iwasawa invariants associated to multigraphs

TL;DR

This paper explores the behavior of Iwasawa invariants in abelian $ ext{ell}$-towers of multigraphs, analyzing their growth patterns and statistical properties within a graph-theoretic Iwasawa theory framework.

Contribution

It introduces a systematic study of Iwasawa invariants for multigraphs, extending classical Iwasawa theory concepts to a graph setting and formulating related statistical questions.

Findings

Formulated statistical questions about Iwasawa invariants in multigraphs.

Analyzed growth patterns of spanning trees in abelian $ ext{ell}$-towers.

Extended Iwasawa theory concepts to multigraphs.

Abstract

Let $\ell$ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell$-towers of multigraphs. In this context, growth patterns are realized by certain analogues of Iwasawa invariants, which depend on the prime $\ell$ and the abelian $\ell$-tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa $\mu$ and $\lambda$ invariants.