Let $\Gamma$ be a distance-regular graph of diameter 3 with eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3$. If $\theta_2=-1$, then the graph $\Gamma_3$ is strongly regular and the complementary graph $\bar\Gamma_3$ is pseudogeometric for $pG_{c_3}(k,b_1/c_2)$. If $\Gamma_3$ does not contain triangles and the number of its vertices~$v$ is less than 800, then $\Gamma$ has intersection array {69,56,10;1,14,60}. In this case $\Gamma_3$ is a graph with parameters (392,46,0,6) and $\bar \Gamma_2$ is a strongly regular graph with parameters (392,115,18,40). Note that the neighborhood of any vertex in a graph with parameters (392,115,18,40) is a strongly regular graph with parameters (115,18,1,3), and its existence is unknown. In this paper, we find possible automorphisms of this strongly regular graph and automorphisms of a distance-regular graph with intersection array {69,56,10;1,14,60}. In particular, it is proved that the latter graph is not arc-transitive.
Keywords: distance-regular graph, automorphism of a graph.
The paper was received by the Editorial Office on February 27, 2017.
Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: makhnev@imm.uran.ru
Marina Sefovna Nirova, Cand. Phys.-Math. Sci, Kabardino-Balkarian State University named after H. M. Berbekov, Nal’chik, 360004 Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: nirova_m@mail.ru
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