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: s