ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

Abstract

In the present paper, we classify abelian antipodal distance-regular graphs \(\Gamma\) of diameter 3 with the following property: \((*)\) \(\Gamma\) has a transitive group of automorphisms \(\widetilde{G}\) that induces a primitive almost simple permutation group \(\widetilde{G}^{\Sigma}\) on the set \({\Sigma}\) of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank \({\rm rk}(\widetilde{G}^{\Sigma})\) of \(\widetilde{G}^{\Sigma}\) equals 2 moreover, all such graphs are now known. Here we focus on the case \({\rm rk}(\widetilde{G}^{\Sigma})=3\).Under this condition the socle of \(\widetilde{G}^{\Sigma}\) 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 \(\Gamma\) with the property \((*)\) such that \(rk(\widetilde{G}^{\Sigma})=3\) and the socle of \(\widetilde{G}^{\Sigma}\) 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 \(\widetilde{G}^{\Sigma}\). We follow a classification scheme that is based on a reduction to minimal quotients of \(\Gamma\) that inherit the property  \((*)\). For each given group \(\widetilde{G}^{\Sigma}\) with simple classical socle of degree \(|{\Sigma}|\le 2500\), we determine potential minimal quotients of \(\Gamma\), 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 \(\Gamma\) in the case of classical socle for \(\widetilde{G}^{\Sigma}\) under condition \(|{\Sigma}|\le 2500.\)