A distance-transitive graph (DTG) 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 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. The paper gives a complete answer to a problem posed by a world-class scientist M. Guidici during a workshop in 2008 in Banff, Canada. More specifically, it is shown that every DTG admits a semiregular automorphism.

COBISS.SI-ID: 1024085332

Bannai and Ito defined association scheme theory as doing ''group theory without groups'', thus raising a basic question as to which results about permutation groups are, in fact, results about association schemes? By considering transitive permutation groups in a wider setting of association schemes, it is shown that one such result is a generalisation from odd primes p to arbitrary prime powers pn, of the classical theorem of Wielandt about primitive permutation groups of degree 2p, p > 2 a prime, being of rank 3.

COBISS.SI-ID: 1024198996