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arXiv · 2102.04239

On a representation of the automorphism group of a graph in a unimodular group

István Estelyi·Ján Karabáš·Alexander Mednykh·Roman Nedela
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Abstract

We investigate a representation of the automorphism group of a connected graph $X$ in the group of unimodular matrices $U_\beta$ of dimension $\beta$, where $\beta$ is the Betti number of graph $X$. We classify the graphs for which the automorphism group does not embed into $U_\beta$. It follows that if $X$ has no pendant vertices and $X$ is not a simple cycle, then the representation is faithful and $\mathrm{Aut}\,X$ acts faithfully on $H_1(X,\mathbb{Z})$. The latter statement can be viewed as a discrete analogue of a classical Hurwitz's theorem on Riemann surfaces of genera greater than one.

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István Estelyi, Ján Karabáš, Alexander Mednykh, Roman Nedela. 2021-02-08. On a representation of the automorphism group of a graph in a unimodular group. https://arxiv.org/abs/2102.04239

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