  Given a graph and a representation of its fundamental group, there is a
naturally associated twisted adjacency operator. The main result of this
article is the fact that these operators behave in a controlled way under graph
covering maps. When such an operator can be used to enumerate objects, or
compute a partition function, this has concrete implications on the
corresponding enumeration problem, or statistical mechanics model. For example,
we show that if $\widetilde{\Gamma}$ is a finite connected covering graph of a
graph $\Gamma$ endowed with edge-weights $x=\{x_e\}_e$, then the spanning tree
partition function of $\Gamma$ divides the one of $\widetilde{\Gamma}$ in the
ring $\mathbb{Z}[x]$. Several other consequences are obtained, some known,
others new.
