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.
Categories: math.CO math-ph math.MP