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.
Zhao et al. (Wed,) studied this question.