Covers of complete graphs and related association schemes

Abstract

We investigate the association schemes Inv(G) that are formed by the collection of orbitals of a permutation group G, for which the (underlying) graph Γ of a basis relation is a distance-regular antipodal cover of the complete graph. The group G can be regarded as an edge-transitive group of automorphisms of Γ and induces a 2-homogeneous permutation group on the set of its antipodal classes, which is either almost simple and 2-transitive, or affine. Using the classification of the finite 2-transitive permutation groups, we determine every possibility for Γ except for some special cases. This allows us to obtain a classification of such schemes Inv(G) provided that G is quasi-simple. We also give a general characterization of edge-transitive distance-regular antipodal covers of complete graphs in the almost simple case in terms of graphs of basis relations of schemes Inv(G) with quasi-simple group G. Then, we find several constructions of Deza graphs that are associated with such a scheme Inv(G). Finally, we establish isomorphisms between some graphs of basis relations of Inv(G) with G≃SU3(q) and abelian covers related to generalised quadrangles.