Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field , we examine a family of graphs, defined for each pair of integers , with vertex set formed by all injective linear transformations and edges corresponding to pairs of mappings, and , with . For , this graph will be denoted by . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each for . Using the properties of line-transitive groups, we completely determine which of the graphs are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs , and , with and arbitrary, and of two sporadic examples and . Hence, the overwhelming majority of our graphs is not Cayley.