TY - UNPB
T1 - Spanning trees in Z-covers of a finite graph and Mahler measures
AU - Pengo, Riccardo
AU - ValliÃ¨res, Daniel
N1 - Comments are very welcome!
PY - 2023/10/24
Y1 - 2023/10/24
N2 - 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.
AB - 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.
KW - math.NT
KW - math.CO
KW - 05C25, 11B83, 11R06
U2 - 10.48550/arXiv.2310.15619
DO - 10.48550/arXiv.2310.15619
M3 - Preprint
BT - Spanning trees in Z-covers of a finite graph and Mahler measures
ER -