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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.
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Preprints, Working Papers, ...
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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 : Wednesday, January 6, 2021 - 3:28:19 AM

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  • HAL Id : hal-03088070, version 1

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

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