Automorphisms of a distance regular graph with intersection array $\{48,35,9;1,7,40\}$

A. A. Makhnev ; V. V. Bitkina ; A. K. Gutnova

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 2, pp. 24-33

Résumé

If a distance-regular graph $\Gamma$ of diameter $3$ contains a maximal locally regular $1$-code perfect with respect to the last neighborhood, then $\Gamma$ has an intersection array $\{a(p+1),cp,a+1;1,c,ap\}$ or ${\{a(p+1),(a+1)p,c;1,c,ap\}}$, where $a=a_3$, $c=c_2$, $p=p^3_{33}$ (Jurisic and Vidali). In the first case, $\Gamma$ has an eigenvalue $\theta_2=-1$ and $\Gamma_3$ is a pseudo-geometric graph for $GQ(p+1,a)$. If $c=a-1=q$, $p=q-2$, then $\Gamma$ has an intersection array $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$, $q>6$. The orders and subgraphs of fixed points of automorphisms of a hypothetical distance-regular graph with intersection array $\{48,35,9;1,7,40\}$ ($q=7$) are studied in the paper. Let $G={\rm Aut} (\Gamma)$ be an insoluble group acting transitively on the set of vertices of the graph $\Gamma$, $K=O_7(G)$, $\bar T$ be the socle of the group $\bar G=G/K$. Then $\bar T$ contains the only component $\bar L$, $\bar L$ that acts exactly on $K$, $\bar L\cong L_2(7),A_5,A_6,PSp_4(3)$ and for the full the inverse image of $L$ of the group $\bar L$ we have $L_a=K_a\times O_{7'}(L_a)$ and $|K|=7^3$ in the case of $\bar L\cong L_2(7)$, $|K|=7^4$ otherwise.