Iwasawa theory for vertex-weighted graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908369255661568 |
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| author | Murooka, Ryosuke Tateno, Sohei |
| author_facet | Murooka, Ryosuke Tateno, Sohei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iwasawa theory for vertex-weighted graphs Murooka, Ryosuke Tateno, Sohei Combinatorics Number Theory 5C22, 11R23, 5C25 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. |
| title | Iwasawa theory for vertex-weighted graphs |
| topic | Combinatorics Number Theory 5C22, 11R23, 5C25 |
| url | https://arxiv.org/abs/2505.12351 |