Geodesic
On $G$-vertex-transitive covers of complete graphs having at most two $G$-orbits on the arc set
Ural mathematical journal, Tome 10 (2024) no. 1, pp. 147-158 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate abelian (in the sense of Godsil and Hensel) distance-regular covers of complete graphs with the following property: there is a vertex-transitive group of automorphisms of the cover which possesses at most two orbits in the induced action on its arc set. We focus on covers whose parameters belong to some known infinite series of feasible parameters. We also complete the classification of arc-transitive covers with a non-solvable automorphism group and show that the automorphism group of any unknown edge-transitive cover induces a one-dimensional affine permutation group on the set of its antipodal classes.
Keywords: Distance-regular graph, Vertex-transitive graph, Arc-transitive graph
Mots-clés : Antipodal cover 
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Ludmila Yu. Tsiovkina. On $G$-vertex-transitive covers of complete graphs having at most two $G$-orbits on the arc set. Ural mathematical journal, Tome 10 (2024) no. 1, pp. 147-158. http://geodesic.mathdoc.fr/item/UMJ_2024_10_1_a12/

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

[2] Berggren J. L., “An algebraic characterization of finite symmetric tournaments”, Bull. Austral. Math. Soc., 6:1 (1972), 53–59 | DOI | MR | Zbl

[3] Brouwer A. E., Cohen A. M., Neumaier A., Distance–Regular Graphs, Springer–Verlag, Berlin etc., 1989, 494 pp. | DOI | MR | Zbl

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

[5] Chen G., Ponomarenko I., Lecture Notes on Coherent Configurations, 2023, 356 pp. https://www.pdmi.ras.ru/~ inp/ccNOTES.pdf

[6] Coutinho G., Godsil C., Shirazi M., Zhan H., “Equiangular lines and covers of the complete graph”, Lin. Alg. Appl., 488 (2016), 264–283 | DOI | MR | Zbl

[7] Coutts H. J., Quick M. R., Roney-Dougal C. M., “The primitive groups of degree less than $4096$”, Comm. Algebra, 39 (2011), 3526–3546 | DOI | MR | Zbl

[8] Devillers A., Giudici M., Li C.H., Pearce G., Praeger Ch. E., “On imprimitive rank 3 permutation groups”, J. London Math. Soc., 84 (2011), 649–669 | DOI | MR | Zbl

[9] Godsil C. D., Hensel A. D., “Distance regular covers of the complete graph”, J. Comb. Theory Ser. B., 56:1 (1992), 205–238 | DOI | MR | Zbl

[10] 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 | MR | Zbl

[11] Jones W., Parshall B., “On the $1$-cohomology of finite groups of Lie type”, Proc. Conf. on Finite Groups, eds. W.R. Scott, F. Gross, Academic Press, New York, 1976, 313–327 | DOI | MR

[12] Tsiovkina L. Yu., “Covers of complete graphs and related association schemes”, J. Comb. Theory Ser. A., 191 (2022), 105646 | DOI | MR | Zbl

[13] Tsiovkina L. Yu., “On a class of edge-transitive distance-regular antipodal covers of complete graphs”, Ural Math. J., 7:2 (2021), 136–158 | DOI | MR | Zbl

[14] Tsiovkina L. Yu., “Arc-transitive groups of automorphisms of antipodal distance-regular graphs of diameter 3 in affine case”, Sib. Èlektron. Mat. Izv., 17 (2020), 445–495 (in Russian) | DOI | MR | Zbl

[15] Tsiovkina L. Yu., “On a class of vertex-transitive distance-regular covers of complete graphs”, Sib. Èlektron. Mat. Izv., 18:2 (2021), 758–781 (in Russian) | DOI | MR | Zbl

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

[17] Tsiovkina L. Yu., “On some vertex-transitive distance-regular antipodal covers of complete graphs”, Ural Math. J., 8:2 (2022), 162–176 | DOI | MR | Zbl

[18] Wilson R. A., The Finite Simple Groups, v. 251, Grad. Texts in Math., Springer—Verlag, London, 2009, 298 pp. | DOI | MR | Zbl