Iwasawa theory for vertex-weighted graphs

TL;DR

The paper develops a vertex-weighted analogue of arithmetic Iwasawa theory for finite graphs by extending the matrix-tree theorem to field-valued weights, introducing voltage assignments and $h$-functions, and proving a decomposition formula for derived graphs. It then establishes Iwasawa-type growth formulas for $\\mathbb{Z}_p^d$-towers of vertex-weighted graphs and a vertex-weighted Kida's formula, providing explicit invariant-based descriptions of rooted and non-rooted weighted complexities. Building on and refining prior work (Chung–Langlands, Wu–Feng–Sato, Gonet, Vallières), it unifies combinatorial and arithmetic perspectives with rigorous asymptotics and numerical demonstrations. The results offer a combinatorial framework and precise growth estimates that mirror arithmetic Iwasawa theory, enabling sharp root-wise complexity growth analysis in weighted graph towers.

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 Vallières 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.

Figures (7)

• Figure 2.1: symmetric directed graph $X$
• Figure 2.2: five spanning trees of $X$
• Figure 2.3: four rooted trees of $T_1$
• Figure 2.4: The graph $X$
• Figure 2.5: the derived graph $X(\alpha)$

Theorems & Definitions (81)

• Theorem 1.1: = Corollary \ref{['coro_1']}
• Theorem 1.2: = Theorem \ref{['thm_5']}
• Theorem 1.3: = \ref{['eq_14']} in Theorem \ref{['thm_6']}
• Theorem 1.4: = Theorem \ref{['thm_6']} \ref{['eq_21']}
• Theorem 1.5: AMT
• Theorem 1.6: = Theorem \ref{['thm_2']}
• Theorem 1.7: = Theorem \ref{['thm_3']}
• Theorem 1.8: = Theorem \ref{['thm_40']}
• Definition 2.1
• Remark 2.2