  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.
