Automorphisms of a Distance-Regular Graph with Intersection Array $\{30,22,9;1,3,20\}$

Abstract

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 $\Gamma$, $G=\mathrm{Aut}(\Gamma)$ 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 $F_{2}U_{3}(5)$-module $V$ by $U_{3}(5)$ and the dimension of $V$ over $F_{3}$ is $20$, $28$, $56$, $104$, or $288$.