Spanning Trees in ℤ-Covers of a finite Graph and Mahler Measures

authored by

Riccardo Pengo, Daniel Vallières

Abstract

Using the special value at of Artin-Ihara L-functions, we associate to every -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular -cover, our result gives us back Lengyel's calculation of the p-adic valuations of Fibonacci numbers.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

The California State University

Type

Article

Journal

Journal of the Australian Mathematical Society

Volume

118

Pages

108-144

No. of pages

37

ISSN

1446-7887

Publication date

02.2025

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics

Electronic version(s)

https://doi.org/10.1017/S1446788724000144 (Access: Closed)
https://doi.org/10.48550/arXiv.2310.15619 (Access: Open)