On Some Vertex-Transitive Distance-Regular Antipodal Covers of Complete Graphs

Abstract

In the present paper, we classify abelian antipodal distance-regular graphs Γ of diameter 3 with the following property: (*) Γ has a transitive group of automorphisms that induces a primitive almost simple permutation group Σ on the set Σ of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank rk(Σ) of Σ equals 2; moreover, all such graphs are now known. Here we focus on the case rk(Σ) = 3. Under this condition the socle of Σ turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs Γ with the property (*) such that rk(Σ) = 3 and the socle of Σ is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for Σ. We follow a classification scheme that is based on a reduction to minimal quotients of Γ that inherit the property (*). For each given group Σ with simple classical socle of degree |Σ|≤ 2500, we determine potential minimal quotients of Γ, applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of Γ in the case of classical socle for Σ under condition |Σ|≤ 2500.