Spanning trees and their relations in Galois covers
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913757337223168 |
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| author | Mizuno, Kosuke |
| author_facet | Mizuno, Kosuke |
| contents | This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^m$-covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin--Ihara $L$-function, we prove two formulas which are graph-theoretic analogues of Kuroda's formula and the Brauer--Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spanning trees and their relations in Galois covers Mizuno, Kosuke Combinatorics Number Theory 5C25, 11R29, 11R32 This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^m$-covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin--Ihara $L$-function, we prove two formulas which are graph-theoretic analogues of Kuroda's formula and the Brauer--Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic. |
| title | Spanning trees and their relations in Galois covers |
| topic | Combinatorics Number Theory 5C25, 11R29, 11R32 |
| url | https://arxiv.org/abs/2503.19641 |