A graph G is distance transitive if its automorphism group is transitive on pairs of vertices at each pairwise distance in the graph. Distance-transitivity is a generalization of distance-regularity. Every distance-transitive graph is distance-regular, but the converse does not necessarily hold, as first shown by Adel'son-Vel'skii et al. (1969; Brouwer et al. 1989, p. 136). The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph . While it is most common to consider only connected distance-transitive graphs, the above definition applies equally well to disconnected graphs, where in addition to integer graph distances, pairs of vertices in different connected components are considered to be at distance infinity.
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