Finite Vertex Bi-Primitive 2-Arc Transitive Graphs Admitting a Two-Dimensional Linear Group

A graph is said to be vertex bi-primitive, if it is a bipartite graph, and the setwise stabilizer of its automorphism group acts primitively on two bi-parts. In this paper, we not only classify vertex bi-primitive 2-arc transitive graphs admitting a two-dimensional linear group but also determine the automorphism groups and the number of non-isomorphic ones of such graphs.