Spanning trees in Z-covers of a finite graph and Mahler measures

Research output: Working paper/Preprint › Preprint

Authors

  • Riccardo Pengo
  • Daniel Vallières

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Original languageEnglish
Publication statusE-pub ahead of print - 24 Oct 2023

Abstract

Using the special value at \(u=1\) of Artin-Ihara \(L\)-functions, we associate to every \(\mathbb{Z}\)-cover of a finite 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. Specializing 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 generalized 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 \(\mathbb{Z}\)-cover, our result gives us back Lengyel's calculation of the \(p\)-adic valuations of Fibonacci numbers.

Keywords

    math.NT, math.CO, 05C25, 11B83, 11R06

Cite this

Spanning trees in Z-covers of a finite graph and Mahler measures. / Pengo, Riccardo; Vallières, Daniel.
2023.

Research output: Working paper/Preprint › Preprint

Pengo R, Vallières D. Spanning trees in Z-covers of a finite graph and Mahler measures. 2023 Oct 24. Epub 2023 Oct 24. doi: 10.48550/arXiv.2310.15619
Pengo, Riccardo ; Vallières, Daniel. / Spanning trees in Z-covers of a finite graph and Mahler measures. 2023.
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