  Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of
the number of spanning trees in an abelian $\ell$-tower of connected
multigraphs. This is analogous to the classical theorem of Washington--Sinnott
on the growth of the $p$-part of the class group in a cyclotomic
$\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore,
we show that under certain hypotheses, the number of primes dividing the number
of spanning trees is unbounded in such a tower.
