On Unimodular Matrix Representation of the Automorphism Group of a Graph

Recently, Estélyi et al. investigated a representation [Formula: see text] of the automorphisms of a connected graph [Formula: see text] by [Formula: see text] unimodular matrices over [Formula: see text], where [Formula: see text] is the Betti number of [Formula: see text], and classified the graphs for which the representation is unfaithful, with two problems left open: (1) What is the smallest dimension [Formula: see text] such that [Formula: see text] is faithfully represented by [Formula: see text] unimodular matrices? (2) Given a finite group [Formula: see text], find the smallest graph [Formula: see text] such that [Formula: see text] is faithfully represented by unimodular matrices. In this article, we prove that [Formula: see text] has a faithful representation by unimodular matrices of dimension [Formula: see text] if every major vertex of [Formula: see text] has at least four subneighbors. We also classify the graphs [Formula: see text] with Betti number [Formula: see text] for which the image of [Formula: see text] under [Formula: see text] has a subgroup isomorphic to the symmetry group on [Formula: see text] elements.