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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2022, Volume 19, Issue 1, Pages 348–359
DOI: https://doi.org/10.33048/semi.2022.19.030
(Mi semr1506)
 

This article is cited in 1 scientific paper (total in 1 paper)

Discrete mathematics and mathematical cybernetics

On a class of vertex-transitive distance-regular covers of complete graphs, II

L. Yu. Tsiovkina

Krasovsky Institute of Mathematics and Mechanics, 16, S. Kovalevskoi str., Yekaterinburg, 620090, Russia
Full-text PDF (425 kB) Citations (1)
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Abstract: Let $\Gamma$ be an abelian antipodal distance-regular graph of diameter 3 with the following property: $(*)$ $\Gamma$ has a transitive group $\overline{G}$ of automorphisms which induces a primitive almost simple permutation group $\overline{G}^{\Sigma}$ on the set ${\Sigma}$ of its antipodal classes. If permutation rank ${\rm rk}(\overline{G}^{\Sigma})$ of $\overline{G}^{\Sigma}$ equals $2$, then $\Gamma$ is arc-transitive; moreover, all such graphs are now known. The purpose of this paper is to describe the graphs $\Gamma$ with the property $(*)$ in the case when ${\rm rk}(\overline{G}^{\Sigma})=3$. According to the classification of primitive almost simple permutation groups of rank $3$ the socle of the group $\overline{G}^{\Sigma}$ under the given condition is either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, we described the graphs $\Gamma$ provided that ${\rm rk}(\overline{G}^{\Sigma})=3$ and the socle of $\overline{G}^{\Sigma}$ is a sporadic simple group. Here we study the cases when $(i)$ the socle of the group $\overline{G}^{\Sigma}$ is an alternating group or $(ii)$ $|{\Sigma}|\le 2500$ and socle of $\overline{G}^{\Sigma}$ is a simple group of exceptional Lie type. We show that the family of non-bipartite graphs $\Gamma$ with the property $(*)$ and $\mathrm{rk}(\overline{G}^{\Sigma})=3$ in the alternating case is finite and limited to a small number of potential examples with $|\Sigma|\in\{10,28,120\}$, each of which is a covering of one of five certain distance-transitive Taylor graphs. For each given group $\overline{G}^{\Sigma}$ of degree $|{\Sigma}|\le 2500$ of exceptional type, we essentially restrict the set of admissible parameters of $\Gamma$.
Keywords: distance-regular graph, antipodal cover, abelian cover, vertex-transitive graph, rank 3 group.
Funding agency Grant number
Russian Science Foundation 20-71-00122
Received March 25, 2022, published July 5, 2022
Bibliographic databases:
Document Type: Article
UDC: 512.542.7, 519.17
MSC: 05B25, 05E18
Language: Russian
Citation: L. Yu. Tsiovkina, “On a class of vertex-transitive distance-regular covers of complete graphs, II”, Sib. Èlektron. Mat. Izv., 19:1 (2022), 348–359
Citation in format AMSBIB
\Bibitem{Tsi22}
\by L.~Yu.~Tsiovkina
\paper On a class of vertex-transitive distance-regular covers of complete graphs, II
\jour Sib. \`Elektron. Mat. Izv.
\yr 2022
\vol 19
\issue 1
\pages 348--359
\mathnet{http://mi.mathnet.ru/semr1506}
\crossref{https://doi.org/10.33048/semi.2022.19.030}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4449222}
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  • https://www.mathnet.ru/eng/semr/v19/i1/p348
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