Almost rank three graphs (Q1195476)

A finite graph is pair-symmetric if its automorphism group acts transitively on four sets of ordered pairs of vertices: edges which lie in a triangle or in no triangle and non-edges which lie in a cotriangle or in no cotriangle. The author obtains necessary and sufficient conditions for a graph to be pair-symmetric and shows among other things that the automorphism group of a pair-symmetric graph nearly always has rank at most three.