A distance-regular graph $\Gamma$ with intersection array $\{115,96,30,1;1,10,96,175\}$ is an $AT4$-graph. The antipodal quotient $\bar \Gamma$ 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.
The paper was received by the Editorial Office on April 7, 2017
Archil Khazeshovich Zhurtov, Dr. Phys.-Math. Sci., Kabardino-Balkarian State University named after H.M. Berbekov, Nal’chik, 360004 Russia, e-mail: zhurtov_a@mail.ru
Mariyana Khusenovna Shermetova, doctoral student, Kabardino-Balkarian State University named afterH.M. Berbekov, Nal’chik, 360004 Russia, e-mail: mariyana1992@mail.ru
REFERENCES
1. Brouwer A.E., Cohen A.M., Neumaier A. Distance-Regular Graphs. Berlin etc: Springer-Verlag, 1989. 495 p. ISBN: 3-540-50619-5.
2. Makhnev A.A., Paduchikh D.V. Small $AT4$-graphs and strongly regular subgraphs corresponding to them. Proc. Steklov Institute Math., 2017, vol. 296, no. 1, pp. 164-174. S1-S11. doi: 10.1134/S0081543817020158.
3. Soicher L.H. Uniqueness of a distance-regular graph with intersection array $\{32,27,8,1;1,4,27,32\}$ and related results [e-resource]. 2015. 11 c. Available at: https://arxiv.org/pdf/1512.05976.pdf.
4. Nirova M.C. Automorphisms of a distance-regular graph with intersection array $\{144,125,32,1;1,8,125,144\}$. Siberian. Electronic Mathematical Reports, 2017, vol. 14, pp. 178-189. doi: 10.17377/semi.2017.14.018.
5. Makhnev A.A., Paduchikh D.V., Samoilenko M.S. Automorphisms of a graph with intersection array $\{115,96,30,1;1,10,96,115\}$. Dokl. Math., 2014, vol. 90, no. 3, pp. 692-696. doi: 10.1134/S1064562414060131.
6. Makhnev A.A., Nirova M.S. Automorphisms of a distance-regular graph with intersection array $\{69,56,10;1,14,60\}$. Trudy Inst. Mat. Mekh. UrO RAN, 2017, vol. 23, no. 3, pp. 182-190 (in Russian). doi: 10.21538/0134-4889-2017-23-3-182-190.
7. Makhnev A.A., Ponomarev D.N. Automorphisms of a strongly regular graph with parameters (392,115,18,40). Dokl. Math., 2015, vol. 91, no. 1, pp. 12-15. doi: 10.1134/S1064562414070035.
8. Makhnev A.A., Samoilenko M.S. Automorphisms of a strongly regular graph with parameters (276,75,10,24). Dokl. Math., 2014, vol. 90, no. 1, pp. 485-488. doi: 10.1134/S1064562414050238.
9. Cameron P.J. Permutation Groups. Cambridge: Cambridge Univ. Press. 1999, Ser. London Math. Soc. Student Texts, vol. 45, 232 p. ISBN: 0-521-65302-9.
10. Makhnev A.A., Paduchikh D.V., Tsiovkina L.Yu. Antipodal distance-regular covers of Hermitian form graphs $Herm(2,q^2)$. Dokl. Math., 2015, vol. 91, no. 3, pp. 304-308. doi: 10.1134/S106456241503014X.
11. Gavrilyuk A.L., Makhnev A.A., On automorphisms of distance-regular graph with the intersection array $\{56,45,1;1,9,56\}$. Dokl. Math., 2010, vol. 81, no. 3, pp. 439-442. doi: 10.1134/S1064562410030282.
12. Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Siberian. Electr. Math. Reports, 2009, vol. 6, pp. 1-12.
