The non-$$\ell $$-part of the number of spanning trees in abelian $$\ell $$-towers of multigraphs

Abstract

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.