Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

Abstract

Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{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.