We investigate a representation of the automorphism group of a connected graph $X$ in the group of unimodular matrices $U_β$ of dimension $β$, where $β$ is the Betti number of graph $X$. We classify the graphs for which the automorphism group does not embed into $U_β$. 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.