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 $\tilde{\Gamma }$ 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 $\tilde{\Gamma }$ in the ring $\mathbb{Z}[x]$. Several other consequences are obtained, some known, others new.

several changes in v2, including a shorter proof of the main result; to appear in ALCO; v2 also contains an addendum not included in the version in press

References in corpus (1)

• The dimer and Ising models on Klein bottles

Cited by in corpus (1)

• A variational principle for domino tilings of multiply-connected domains