  We study the special value $u=1$ of Artin-Ihara $L$-functions associated to
characters of the automorphism group of abelian covers of multigraphs. In
particular, we show an annihilation statement analogous to a classical
conjecture of Brumer on annihilation of class groups for abelian extensions of
number fields and we also calculate the index of an ideal analogous to the
classical Stickelberger ideal in algebraic number theory. Along the way, we
make some observations about the number of spanning trees in abelian multigraph
coverings that may be of independent interest.
