New interface

# 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 $\widetilde{\Gamma}$ 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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03088070
Contributor : Adrien Kassel Connect in order to contact the contributor
Submitted on : Friday, December 25, 2020 - 8:52:48 AM
Last modification on : Friday, September 30, 2022 - 11:26:51 AM

### Files

graph-coverings.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-03088070, version 1

### Citation

David Cimasoni, Adrien Kassel. Graph coverings and twisted operators. 2020. ⟨hal-03088070⟩

Record views