Iwasawa theory for vertex-weighted graphs

2025/05/18 by Ryosuke Murooka, Sohei Tateno, Murooka, Ryosuke +1
Mathematics · #Analytic Number Theory Research #Commutative Algebra and Its Applications #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2505.12351

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 a refinement of Kida's formula for 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 ℤ_p^d-towers. We also provide several numerical examples.

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