On abelian \(\ell\)-towers of multigraphs III

Let \(\ell\) be a rational prime. Previously, abelian \(\ell\)-towers of multigraphs were introduced which are analogous to \(\Z_{\ell}\)-extensions of number fields. It was shown that for towers of bouquets, the growth of the \(\ell\)-part of the number of spanning trees behaves in a predictable ma...

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Bibliographic Details
Published in:arXiv.org 2021-07
Main Authors: McGown, Kevin J, Vallières, Daniel
Format: Article
Language:English
Subjects:
Graph theory
Number theory
Polynomials
Power series
Towers
Online Access:Get full text
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Summary:Let \(\ell\) be a rational prime. Previously, abelian \(\ell\)-towers of multigraphs were introduced which are analogous to \(\Z_{\ell}\)-extensions of number fields. It was shown that for towers of bouquets, the growth of the \(\ell\)-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for \(\Z_{\ell}\)-extensions of number fields). In this paper, we extend this result to abelian \(\ell\)-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in \(\Z_\ell\) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at \(u=1\) of the Artin--Ihara \(L\)-function, when the base multigraph is not necessarily a bouquet.
ISSN:2331-8422