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arXiv · 2012.12575

Graph coverings and twisted operators

David Cimasoni·Adrien Kassel
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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 $\widetildeΓ$ is a finite connected covering graph of a graph $Γ$ endowed with edge-weights $x=\{x_e\}_e$, then the spanning tree partition function of $Γ$ divides the one of $\widetildeΓ$ in the ring $\mathbb{Z}[x]$. Several other consequences are obtained, some known, others new.

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David Cimasoni, Adrien Kassel. 2023-02-08. Graph coverings and twisted operators. https://doi.org/10.5802/alco.258

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