Iwasawa theory for vertex-weighted graphs

TL;DR

This paper extends Iwasawa theory to vertex-weighted graphs, generalizing matrix-tree theorems and zeta functions, and provides formulas for growth in graph towers with practical examples.

Contribution

It introduces a generalized Iwasawa theory for vertex-weighted graphs, including new formulas and decomposition theorems applicable over arbitrary fields.

Findings

Generalized matrix-tree theorem for field-valued vertex-weighted graphs

Proved Iwasawa-type and Kida's formulas for graph towers

Estimated root-wise growth of weighted complexities

Abstract

Chung-Langlands established a matrix-tree theorem for positive-real valued vertex-weighted graphs, and Wu-Feng-Sato developed a theory of Ihara zeta functions for those graphs. In this paper, generalizing and refining these previous works, we initiate the Iwasawa theory for vertex-weighted graphs, which is a generalization of the Iwasawa theory for graphs initiated by Gonet and Valli\`{e}res independently. First, we generalize the matrix-tree theorem by Chung-Langlands to arbitrary field-valued vertex-weighted graphs. Second, we refine and prove the so-called decomposition formula for vertex-weighted graphs and edge-weighted graphs without any assumption. Applying these results, we prove the Iwasawa-type formula and Kida's formula for $\mathbb{Z}_p^d$-towers of vertex-weighted graphs. Our refinement of the decomposition formulas allows us to estimate the root-wise growth of weighted complexities in $\mathbb{Z}_p^d$-towers. We also provide several numerical examples.