Iwasawa theory for weighted graphs

TL;DR

This paper extends Iwasawa theory to weighted graphs, establishing analogues of key formulas and invariants, and explores applications to quantum walks, providing both theoretical results and numerical examples.

Contribution

It generalizes Iwasawa theory for graphs to weighted graphs, including formulas and invariants, and connects it to quantum walk applications.

Findings

Proved analogue of Iwasawa's class number formula for weighted graphs.

Established Kida's formula for compatible systems of covers.

Provided numerical examples of invariants and characteristic elements.

Abstract

Let $p$ be a prime number and let $d$ be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Valli\`{e}res to weighted graphs. In particular, we prove an analogue of Iwasawa's class number formula and that of Kida's formula for compatible systems of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.