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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
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.
Образец цитирования:
Ludmila Yu. Tsiovkina, “On some vertex-transitive distance-regular antipodal covers of complete graphs”, Ural Math. J., 8:2 (2022), 162–176
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/umj181 https://www.mathnet.ru/rus/umj/v8/i2/p162
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