K.S. Efimov, A.A. Makhnev. Automorphisms of a distance-regular graph with intersection array {30,22,9;1,3,20} ... P. 23-31

A distance-regular graph $\Gamma$ of diameter 3 is called a Shilla graph if it has the second eigenvalue $\theta_1=a_3$. In this case $a=a_3$ divides $k$ and we set $b=b(\Gamma)=k/a$. Koolen and Park obtained the list of intersection arrays for Shilla graphs with $b=3$. There exist graphs with intersection arrays $\{12,10,5;1,1,8\}$ and $\{12,10,3;1,3,8\}$. The nonexistence of graphs with intersection arrays $\{12,10,2;1,2,8\}$, $\{27,20,10;1,2,18\}$, $\{42,30,12;1,6,28\}$, and $\{105,72,24;1,12,70\}$ was proved earlier. In this paper we study the automorphisms of a distance-regular graph $\Gamma$ with intersection array $\{30,22,9;1,3,20\}$, which is a Shilla graph with $b=3$. Assume that $a$ is a vertex of $\Gamma$, $G={\rm Aut}(\Gamma)$ is a nonsolvable group, $\bar G=G/S(G)$, and $\bar T$ is the socle of $\bar G$. Then $\bar T\cong L_2(7),A_7,A_8,$ or $U_3(5)$. If $\Gamma$ is arc-transitive, then $T$ is an extension of an irreducible $F_2U_3(5)$-module $V$ by $U_3(5)$ and the dimension of $V$ over $F_3$ is 20, 28, 56, 104, or 288.

Keywords: Shilla graph, graph automorphism

Received March 2, 2020

Revised May 26, 2020

Accepted June 15, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research – the National Natural Science Foundation of China (project no. 20-51-53013_a).

Konstantin Sergeevich Efimov, Cand. Sci. (Phys.-Math.), Ural State University of Economics, Yekaterinburg, 620144 Russia, e-mail: konstantin.s.efimov@gmail.com

Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci., RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: makhnev@imm.uran.ru

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Cite this article as: K.S. Efimov, A.A. Makhnev. Automorphisms of a distance-regular graph with intersection array {30,22,9;1,3,20}, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 23–31.