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Automorphisms of a distance regular graph with intersection array $\{48,35,9;1,7,40\}$
A. A. Makhneva, V. V. Bitkinab, A. K. Gutnovab a N. N. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaja St., Ekaterinburg 620990, Russia
b North Ossetian State University, 44-46 Vatutin St., Vladikavkaz 362025, Russia
Abstract:
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.
Key words:
strongly regular graph, distance-regular graph, automorphism of graph.
Received: 30.03.2020
Citation:
A. A. Makhnev, V. V. Bitkina, A. K. Gutnova, “Automorphisms of a distance regular graph with intersection array $\{48,35,9;1,7,40\}$”, Vladikavkaz. Mat. Zh., 22:2 (2020), 24–33
Linking options:
https://www.mathnet.ru/eng/vmj721 https://www.mathnet.ru/eng/vmj/v22/i2/p24
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Abstract page: | 142 | Full-text PDF : | 30 | References: | 19 |
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