Antipodal distance-transitive covers of complete bipartite graphs

Abstract

This paper completes the classification of antipodal distance-transitive covers of the complete bipartite graphs Kk,k, where k ≥ 3. For such a cover the antipodal blocks must have size r ≤ k. Although the case r = k has already been considered, we give a unified treatment of r ≤ k. We use deep group-theoretic results as well as representation-theoretic data about explicit linear groups and group coset geometries. Apart from the generic examples arising from finite projective spaces, there are three sporadic examples (arising from the outer automorphisms of the symmetric group S6 and of the Mathieu group M12 and one related to non-abelian Singer groups on PG2(4)) and an infinite family having solvable automorphism group (and with parameters r = qb, k = qa, where (qb - 1)gcd(b, q - 1) divides 2a (q - 1) and q is a prime power). © 1997 Academic Press Limited.