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 Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 2, Pages 24–33 (Mi vmj721)

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 $G=Aut (\Gamma)$ 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 20-51-53013

DOI: https://doi.org/10.46698/n0833-6942-7469-t

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UDC: 519.17
MSC: 05C25

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

Citation in format AMSBIB
\Bibitem{MakBitGut20} \by A.~A.~Makhnev, V.~V.~Bitkina, A.~K.~Gutnova \paper Automorphisms of a distance regular graph with intersection array $\{48,35,9;1,7,40\}$ \jour Vladikavkaz. Mat. Zh. \yr 2020 \vol 22 \issue 2 \pages 24--33 \mathnet{http://mi.mathnet.ru/vmj721} \crossref{https://doi.org/10.46698/n0833-6942-7469-t}