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On some vertex-transitive distance-regular antipodal covers of complete graphs
Ludmila Yu. Tsiovkina Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Аннотация:
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
$\mathrm{rk}(\widetilde{G}^{\Sigma})=3$ 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
$|{\Sigma}|\le 2500$.
Ключевые слова:
distance-regular graph, antipodal cover, abelian cover, vertex-transitive graph, rank 3 group.
Язык публикации: английский
DOI:
10.15826/umj.2022.2.014