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Journal Articles Algebraic Combinatorics Year : 2023

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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Dates and versions

hal-03088070 , version 1 (25-12-2020)
hal-03088070 , version 2 (08-02-2023)

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David Cimasoni, Adrien Kassel. Graph coverings and twisted operators. Algebraic Combinatorics, 2023, 6 (1), pp.75-94. ⟨10.5802/alco.258⟩. ⟨hal-03088070v2⟩
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