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

TL;DR

This paper studies a specific matrix representation of a graph's automorphism group related to its Betti number, classifies graphs with non-faithful representations, and draws parallels to classical theorems on Riemann surfaces.

Contribution

It introduces a new representation of automorphism groups in unimodular matrices and classifies graphs where this representation is not faithful.

Findings

Representation is faithful for graphs without pendant vertices that are not cycles.

Automorphism group acts faithfully on the first homology group in these cases.

Provides a discrete analogue of Hurwitz's theorem for Riemann surfaces.

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.