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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 2, Pages 11–17 (Mi vmj612)

On automorphisms of a distance-regular graph with intersection of arrays $\{39,30,4; 1,5,36\}$

A. K. Gutnovaa, A. A. Makhnevb

a North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue $\leq t$ for a given positive integer $t$. This problem is reduced to the description of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with non-principal eigenvalue $t$ for $t =1,2,\ldots$ Let $\Gamma$ be a distance regular graph of diameter $3$ with eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3$. If $\theta_2= -1$, then by Proposition 4.2.17 from the book «Distance-Regular Graphs» (Brouwer A. E., Cohen A. M., Neumaier A.) the graph $\Gamma_3$ is strongly regular and $\Gamma$ is an antipodal graph if and only if $\Gamma_3$ is a coclique. Let $\Gamma$ be a distance-regular graph and the graphs $\Gamma_2$, $\Gamma_3$ are strongly regular. If $k <44$, then $\Gamma$ has an intersection array $\{19,12,5; 1,4,15\}$, $\{35,24,8; 1,6,28\}$ or $\{39,30,4; 1,5,36\}$. In the first two cases the graph does not exist according to the works of Degraer J. «Isomorph-free exhaustive generation algorithms for association schemes» and Jurisic A., Vidali J. «Extremal 1-codes in distance-regular graphs of diameter 3». In this paper we found the possible automorphisms of a distance regular graph with an array of intersections $\{39,30,4; 1,5,36\}$.

Key words: regular graph, symmetric graph, distance-regular graph, automorphism groups of graph.

 Funding Agency Grant Number Russian Science Foundation 15-11-10025 Ministry of Education and Science of the Russian Federation 02.A03.21.0006

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UDC: 519.17

Citation: A. K. Gutnova, A. A. Makhnev, “On automorphisms of a distance-regular graph with intersection of arrays $\{39,30,4; 1,5,36\}$”, Vladikavkaz. Mat. Zh., 19:2 (2017), 11–17

Citation in format AMSBIB
\Bibitem{GutMak17} \by A.~K.~Gutnova, A.~A.~Makhnev \paper On automorphisms of a distance-regular graph with intersection of arrays $\{39,30,4; 1,5,36\}$ \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 2 \pages 11--17 \mathnet{http://mi.mathnet.ru/vmj612}