Geodesic
On a class of vertex-transitive distance-regular covers of complete graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 758-781.

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In this paper, we investigate the problem of classification of abelian antipodal distance-regular graphs $\Gamma$ of diameter three with the following property $(*)$: there is a vertex-transitive group of automorphisms $G$ of $\Gamma$ which induces an almost simple primitive permutation group $G^{\Sigma}$ on the set $\Sigma$ of antipodal classes of $\Gamma$. This problem has been recently solved in the case when the permutation rank $\mathrm{rk}(G^{\Sigma})$ of $G^{\Sigma}$ equals $2$ (which implies classification of all arc-transitive representatives). Here we start to study the next case $\mathrm{rk}(G^{\Sigma})=3$. We elaborate a method of reduction to minimal quotients of $\Gamma$, which gives us a base for a classification scheme that depends on a type of such quotient. By analysing equitable partitions of $\Gamma$ which arise as collections of orbits of some subgroups of $G$, we obtain several strong restrictions on spectra and parameters of $\Gamma$ as well as a description of its minimal quotients. This allows us to settle the case when the socle of $G^{\Sigma}$ is a sporadic simple group.
Keywords: distance-regular graph, abelian cover, vertex-transitive graph, rank $3$ group.
Mots-clés : antipodal cover 
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L. Yu. Tsiovkina. On a class of vertex-transitive distance-regular covers of complete graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 758-781. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a36/

[1] M. Aschbacher, Finite group theory, 2nd ed., Cambridge University Press, Cambridge, 2000 | Zbl

[2] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin etc, 1989 | Zbl

[3] A.E. Brouwer, H. Van Maldeghem, Strongly regular graphs, https://homepages.cwi.nl/ãeb/math/srg/rk3/srgw.pdf

[4] A.E. Brouwer, W.H.Haemers, Spectra of graphs, Springer, Berlin, 2012 | Zbl

[5] J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite groups maximal subgroups and ordinary characters for simple groups, Clarendon Press, Oxford, 1985 (from J. G. Thackray) | Zbl

[6] J.D. Dixon, B. Mortimer, Permutation groups, Springer, New York, 1996 | Zbl

[7] GAP Groups, Algorithms, and Programming, Version 4.7.8, The GAP Group, 2015

[8] A. Gardiner, “Antipodal graphs of diameter three”, Linear Algebra Appl., 46 (1982), 215–219 | DOI | Zbl

[9] A. Gardiner, C.E. Praeger, “On 4-valent symmetric graphs”, Eur. J. Comb., 15:4 (1994), 375–381 | DOI | Zbl

[10] C.D. Godsil, “Krein covers of complete graphs”, Australas. J. Comb., 6 (1992), 245–255 | Zbl

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

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

[13] D. Gorenstein, Finite simple groups. An introduction to their classification, Plenum Press, New York etc, 1982 | Zbl

[14] M.H. Klin, C. Pech, “A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices”, Ars Math. Contemp., 4:2 (2011), 205–243 | DOI | Zbl

[15] V.D. Mazurov, “Minimal permutation representation of Thompson's simple group”, Algebra Logic, 27:5 (1988), 350–361 | DOI | Zbl

[16] A.A. Makhnev, D.V. Paduchikh, L.Yu. Tsiovkina, “Arc-transitive distance-regular coverings of cliques with $\lambda=\mu$”, Proc. Steklov Inst. Math., 284, Suppl. 1 (2014), S124–S134 | DOI | Zbl

[17] A.A. Makhnev, D.V. Paduchikh, L.Yu. Tsiovkina, “Edge-symmetric distance-regular coverings of complete graphs: the almost simple case”, Algebra Logic, 57:2 (2018), 141–152 | DOI | Zbl

[18] B.D. McKay, “Transitive graphs with fewer than twenty vertices”, Math. Comput., 33 (1979), 1101–1121 | DOI | Zbl

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

[20] L.H. Soicher, The GRAPE package for GAP, Version 4.6.1, , 2012 http://www.maths.qmul.ac.uk/l̃eonard/grape/ | Zbl

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

[22] H. Wielandt, Finite permutation groups, Academic Press, New York–London, 1964 | Zbl