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

We investigate a representation of the automorphism group connected graph X in unimodular matrices Uβ dimension β, where β is Betti number X. classify graphs for which does not embed into Uβ. It follows that if has no pendant vertices and simple cycle, then faithful AutX acts faithfully on H1(X,Z). The latter statement can be viewed as discrete analogue classical Hurwitz's theorem Riemann surfaces genera greater than one.