A.A. Makhnev, D.V. Paduchikh, and M.M. Khamgokova gave a classification of distance-regular locally $GQ(5,3)$-graphs. In particular, there arises an $AT4(4,4,2)$-graph with intersection array $\{96,75,16,1;1,16,75,96\}$ on $644$ vertices. The same authors proved that an $AT4(4,4,2)$-graph is not a locally $GQ(5,3)$-graph. However, the existence of an $AT4(4,4,2)$-graph that is a locally pseudo $GQ(5,3)$-graph is unknown. The antipodal quotient of an $AT4(4,4,2)$-graph is a strongly regular graph with parameters $(322,96,20,32)$. These two graphs are locally pseudo $GQ(5,3)$-graphs. We find their possible automorphisms. It turns out that the automorphism group of a distance-regular graph with intersection array $\{96,75,16,1;1,16,75,96\}$ acts intransitively on the set of its antipodal classes.
Keywords: distance-regular graph, graph automorphism.
The paper was received by the Editorial Office on September 1, 2017
Konstantin Sergeevich Yefimov, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620002 Russia;
Ural State University of Economics, 620144 Russia;
Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia;
e-mail: konstantin.s.efimov@gmail.com
REFERENCES
1. Makhnev A.A., Paduchikh D.V., Khamgokova M.M. Dokl. Math., 2010, vol. 82, no. 7, pp. 967-970. doi: 10.1134/S1064562410060335.
2. Jurisic A., Koolen J. Classification of the family $AT4(qs,q,q)$ of antipodal tight graphs. J. Comb. Theory, 2011, vol. 118, no. 3, pp. 842-852. doi: 10.1016/j.jcta.2010.10.001.
3. Makhnev A.A., Paduchikh D.V., Khamgokova M.M. On locally $GQ(5,3)$-graphs. In: Algebra and Combinatorics: Abstracts of the International Conference Dedicated to A.A. Makhnev's 60th Birthday, Ekaterinburg, Russia, 2013 (UMTs-UPI, Ekaterinburg, 2013), pp. 64-66. (in Russian)
4. Brouwer A.E., Haemers W.H. Graph Spectrum. In: Spectra of Graphs. Universitext. Springer, New York. 2012, pp. 1-20. doi: 10.1007/978-1-4614-1939-6_1.
5. Brouwer A.E., Cohen A.M., Neumaier A. Sporadic Graphs. In: Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge A Series of Modern Surveys in Mathematics), vol. 18. Berlin, Heidelberg, Springer. 1989, p.391-412. doi: 10.1007/978-3-642-74341-2_13.
6. Gavrilyuk A.L., Makhnev, A.A. On automorphisms of distance-regular graphs with intersection array $\{56,45,1;1,9,56\}$. Dokl. Math., 2010, vol. 81, no. 3, pp. 439-442. doi: 10.1134/S1064562410030282.
7. Behbahani M., Lam C. Strongly regular graphs with nontrivial automorphisms. Discrete Math., 2011, vol. 311, no. 2-3, pp. 132-144. doi: 10.1016/j.disc.2010.10.005.
8. Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Sibirean electr. math. reports, 2009, vol. 6, pp. 1-12.
