Distance-transitive graphs admit semiregular automorphisms

AbstractA distance-transitive graph is a graph in which for every two ordered pairs of vertices (u,v) and (u′,v′) such that the distance between u and v is equal to the distance between u′ and v′ there exists an automorphism of the graph mapping u to u′ and v to v′. A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism