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

Estélyi, István

On a representation of the automorphism group of a graph in a unimodular group / István Estélyi ... [et al.]. -- 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 Aut X acts faithfully on H_1(X,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.

In Discrete Mathematics. -- Amsterdam : Elsevier B.V., 2021. -- ISSN 0012-365X. -- ISSN 1872-681X. -- Vol. 344, no. 12 (2021), pp. [1-4]