Geodesic
Automorphisms of a distance-regular graph with intersection array $\{75,64,18,1;1,6,64,75\}$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 128-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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A distance-regular graph $\Gamma$ with intersection array $\{115,96,30,1;1,10,96,175\}$ is an $AT4$-graph. The antipodal quotient $\Gamma'$ has parameters $(392,115,18,40)$, and its first and second neighborhoods of vertices are strongly regular with parameters $(115,18,1,3)$ and $(276,75,10,24)$. Moreover, the second neighborhood of any vertex in $\Gamma_2(u)$ has intersection array $\{75,64,18,1;1,6,64,75\}$ and is a $4$-cover of a strongly regular graph with parameters $(276,75,10,24)$. Earlier, Makhnev, Paduchikh, and Samoilenko found possible automorphisms of a graph with parameters $(392,115,18,40)$ and of a graph with intersection array $\{115,96,30,1;1,10,96,175\}$. In this paper we find automorphisms of a graph with intersection array $\{75,64,18,1;1,6,64,75\}$. It is proved that the automorphism group of this graph acts intransitively on the set of its antipodal classes.
Keywords: distance-regular graph, automorphism of a graph. 
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A. Kh. Zhurtov; M. Kh. Shermetova. Automorphisms of a distance-regular graph with intersection array $\{75,64,18,1;1,6,64,75\}$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 128-135. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a12/

[1] Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Springer-Verlag, Berlin etc., 1989, 495 pp. | MR | Zbl

[2] Makhnev A.A., Paduchikh D.V., “Nebolshie $AT4$-grafy i otvechayuschie im silno regulyarnye podgrafy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22:1 (2016), 220–230

[3] Soicher L.H., Uniqueness of a distance-regular graph with intersection array $\{32,27,8,1;1,4,27,32\}$ and related topics, [e-resource], 2015, 11 pp., arXiv: https://arxiv.org/pdf/1512.05976.pdf | MR

[4] Nirova M.S., “Avtomorfizmy distantsionno regulyarnogo grafa s massivami peresechenii $\{144,125,32,1;1,8,125,144\}$”, Sib. elektron. mat. izv., 14 (2017), 178–189 | MR | Zbl

[5] Makhnev A.A., Paduchikh D.V., Samoilenko M.S., “Avtomorfizmy grafa s massivom peresechenii $\{115,96,30,1;1,10,96,115\}$”, Dokl. AN, 459:2 (2014), 149–153 | DOI | MR | Zbl

[6] Makhnev A.A., Nirova M.S., “Ob avtomorfizmakh distantsionno regulyarnogo grafa s massivami peresechenii $\{69,56,10;1,14,60\}$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:3 (2017), 182–190

[7] Makhnev A.A., Ponomarev D.N., “Avtomorfizmy silno regulyarnogo grafa s parametrami (392,115,18,40)”, Dokl. AN, 460:1 (2015), 18–21 | DOI | MR | Zbl

[8] Makhnev A.A., Samoilenko M.S., “Avtomorfizmy silno regulyarnogo grafa s parametrami (276,75,10,24)”, Dokl. AN, 457:5 (2014), 516–519 | DOI | MR | Zbl

[9] Cameron P.J., Permutation groups, London Math. Soc. Student Texts, no. 45, Cambridge Univ. Press, Cambridge, 1999, 232 pp. | MR | Zbl

[10] Makhnev A.A., Paduchikh D.V., Tsiovkina L.Yu., “Distantsionno regulyarnye nakrytiya grafov ermitovykh form $Herm(2,q^2)$”, Dokl. AN, 462:3 (2015), 268–273 | DOI | MR | Zbl

[11] Gavrilyuk A.L., Makhnev A.A., “Ob avtomorfizmakh distantsionno regulyarnogo grafa s massivom peresechenii $\{56,45,1;1,9,56\}$”, Dokl. AN, 432:5 (2010), 583–587 | MR | Zbl

[12] Zavarnitsine A.V., “Finite simple groups with narrow prime spectrum”, Siberian. Electr. Math. Reports, 6 (2009), 1–12 | MR | Zbl