Graph coverings and twisted operators

TL;DR

This paper studies how twisted adjacency operators associated with graph coverings behave, revealing their impact on enumeration problems and statistical mechanics models, including divisibility properties of spanning tree partition functions.

Contribution

It demonstrates that twisted adjacency operators behave predictably under graph coverings, leading to new insights into enumeration and partition functions.

Findings

Spanning tree partition function of a base graph divides that of its finite connected cover.

Behavior of twisted operators under coverings affects enumeration and statistical mechanics models.

Several new consequences for graph invariants and partition functions are established.

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 $\widetilde{\Gamma}$ 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.