Ludmila Yu. Tsiovkina     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)


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 \(|{\Sigma}|\le 2500.\)


Distance-regular graph, Antipodal cover, Abelian cover, Vertex-transitive graph, Rank 3 group

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DOI: http://dx.doi.org/10.15826/umj.2022.2.014

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