Covers of complete graphs and related association schemes. (English) Zbl 1507.05110
Summary: We investigate the association schemes \(\operatorname{Inv}(G)\) that are formed by the collection of orbitals of a permutation group \(G\), for which the (underlying) graph \(\Gamma\) 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 \(\Gamma\) 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 \(\Gamma\) except for some special cases. This allows us to obtain a classification of such schemes \(\operatorname{Inv}(G)\) 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 \(\operatorname{Inv}(G)\) with quasi-simple group \(G\). Then, we find several constructions of Deza graphs that are associated with such a scheme \(\operatorname{Inv}(G)\). Finally, we establish isomorphisms between some graphs of basis relations of \(\operatorname{Inv}(G)\) with \(G \simeq \mathsf{SU}_3(q)\) and abelian covers related to generalised quadrangles.
MSC:
| 05E30 | Association schemes, strongly regular graphs |
| 05C12 | Distance in graphs |
| 20B30 | Symmetric groups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
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