Antipodal Distance-transitive Covers of Complete Bipartite Graphs

This paper completes the classification of antipodal distance-transitive covers of the complete bipartite graphsKk,k,wherek≥3.For such a cover the antipodal blocks must have sizer≤k.Although the caser=khas already been considered, we give a unified treatment ofr≤k.We use deep group-theoretic results as well as representation-theoretic data about explicit linear groups and group coset geometries. from the generic examples arising from finite projective spaces, there are three sporadic examples (arising from the outer automorphisms of the symmetric groupS6and of the Mathieu groupM12and one related to non-abelian Singer groups onPG2(4))and an infinite family having solvable automorphism group (and with parametersr=qb, k=qa,where(qb-1)gcd(b, q-1)divides2a(q-1)andqis a prime power).