On some vertex-transitive distance-regular antipodal covers of complete graphs

Ludmila Yu. Tsiovkina

Ural mathematical journal, Tome 8 (2022) no. 2, pp. 162-176

Résumé

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 $|{\Sigma}|\le 2500$.

Détail

Keywords: distance-regular graph, abelian cover, vertex-transitive graph, rank 3 group.
Mots-clés : antipodal cover

Bibliographie

[1] Aschbacher M., Finite Group Theory, 2-nd ed., Cambridge University Press, Cambridge, 2000, 305 pp. | DOI

[2] Brouwer A.E., Cohen A.M., Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin etc, 1989, 494 pp. | DOI

[3] Brouwer A.E., Van Maldeghem H., Strongly Regular Graphs, Cambridge University Press, Cambridge, 2022, 462 pp. | DOI

[4] Conway J., Curtis R., Norton S., Parker R., Wilson R., Atlas of Finite Groups, Clarendon Press, Oxford, 1985, 252 pp.

[5] Godsil C. D., Hensel A. D., “Distance regular covers of the complete graph”, J. Comb. Theory Ser. B, 56:2 (1992), 205–238 | DOI

[6] Godsil C. D., Liebler R. A., Praeger C. E., “Antipodal distance transitive covers of complete graphs”, Europ. J. Comb., 19:4 (1998), 455–478 | DOI

[7] Gorenstein D., Finite Simple Groups: An Introduction to Their Classification, Springer, New York, 1982 | DOI

[8] Kantor W. M., Liebler R. A., “The rank 3 permutation representations of the finite classical groups”, Trans. Amer. Math. Soc., 271:1 (1982), 1–71 | DOI

[9] Klin M., Pech C., “A new construction of antipodal distance-regular covers of complete graphs through the use of Godsil–Hensel matrices”, Ars Math. Contemp., 4 (2011), 205–243 | DOI

[10] Klin M., Pech C., Reichard S., COCO2P — a GAP4 Package, ver. 0.18, 2020 URL: https://github.com/chpech/COCO2P/

[11] Liebeck M. W., Saxl J., “The finite primitive permutation groups of rank three”, Bull. London Math. Soc., 18:2 (1986), 165–172 | DOI

[12] Mazurov V. D., “Minimal permutation representations of finite simple classical groups. Special linear, symplectic, and unitary groups”, Algebr. Logic, 32:3 (1993), 142–153 | DOI

[13] Roney-Dougal C. M., “The primitive permutation groups of degree less than $2500$”, J. Algebra, 292:1 (2005), 154–183 | DOI

[14] The GAP – Groups, Algorithms, and Programming – a System for Computational Discrete Algebra, ver. 4.7.8, 2015 URL: https://www.gap-system.org/

[15] Soicher L. H., The GRAPE package for GAP, ver. 4.6.1, 2012 URL: https://github.com/gap-packages/grape

[16] Tsiovkina L. Yu., “On a class of vertex-transitive distance-regular covers of complete graphs”, Sib. Elektron. Mat. Izv., 8:2 (2021), 758—781 (in Russian) | DOI

[17] Tsiovkina L. Yu., “On a class of vertex-transitive distance-regular covers of complete graphs II.”, Sib. Electron. Mat. Izv., 19:1 (2022), 348—359 (in Russian) | DOI

[18] Vasil'ev A. V., Mazurov V. D., “Minimal permutation representations of finite simple orthogonal groups”, Algebr. Logic, 33:6 (1995), 337–350 | DOI