  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.
