Details
| Original language | English |
|---|---|
| Pages (from-to) | 108-144 |
| Number of pages | 37 |
| Journal | Journal of the Australian Mathematical Society |
| Volume | 118 |
| Issue number | 1 |
| Early online date | 28 Oct 2024 |
| Publication status | Published - Feb 2025 |
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.
Keywords
- abelian cover of finite graphs, Artin-Ihara L-funtion, Mahler measure, number of spanning trees
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Journal of the Australian Mathematical Society, Vol. 118, No. 1, 02.2025, p. 108-144.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Spanning Trees in ℤ-Covers of a finite Graph and Mahler Measures
AU - Pengo, Riccardo
AU - Vallières, Daniel
N1 - Publisher Copyright: © The Author(s), 2024.
PY - 2025/2
Y1 - 2025/2
N2 - 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.
AB - 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.
KW - abelian cover of finite graphs
KW - Artin-Ihara L-funtion
KW - Mahler measure
KW - number of spanning trees
UR - http://www.scopus.com/inward/record.url?scp=85207941158&partnerID=8YFLogxK
U2 - 10.1017/S1446788724000144
DO - 10.1017/S1446788724000144
M3 - Article
AN - SCOPUS:85207941158
VL - 118
SP - 108
EP - 144
JO - Journal of the Australian Mathematical Society
JF - Journal of the Australian Mathematical Society
SN - 1446-7887
IS - 1
ER -