Distance-transitive action on a graph

A subgroup of the graph automorphism group acts distance-transitively when it is transitive on the ordered pairs of vertices at each fixed graph distance. Thus two vertex pairs at the same distance can be matched by one automorphism from that subgroup. Existence of such a subgroup implies that the full automorphism group has the same property, making the graph a distance-transitive graph. The specified subgroup need not be the entire automorphism group.