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

We investigate a representation of the automorphism group of a connected graph $X$ in the group of unimodular matrices $U_\beta$, of rank $\beta$, where $\beta$ is the Betti number of $X$. We classify the graphs for which the automorphism group does not embed into $U_\beta$. It follows that if $X$ has no pendant vertices, then the representation is faithful, except $X$ is a simple cycle. The latter statement can be viewed as a $1$-dimensional analogue of a classical theorem on Riemann surfaces of genera $>1$.