Graph coverings and twisted operators

Abstract

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 Γ is a finite connected covering graph of a graph Γ endowed with edgeweights x = {xe}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.