Graph coverings and twisted operators

Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator, uniquely defined up to conjugacy. The main result of this article is the fact that this operator behaves 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 Γ ˜ is a finite covering graph of a connected graph Γ endowed with edge-weights x={x e } e , then the spanning tree partition function of Γ divides the one of Γ ˜ in the ring ℤ[x]. Several other consequences are obtained, some known, others new.