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 .
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