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 Vladikavkaz. Mat. Zh., 2018, Volume 20, Number 4, Pages 43–49 (Mi vmj675)

On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$

A. K. Gutnovaa, A. A. Makhnevb

a North Ossetian State University, 44–46 Vatutin Street, Vladikavkaz 362025, Russia
b N.N. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaja st., Ekaterinburg 620990, Russia

Abstract: In the works of A. A. Makhnev and A. K. Gutnova arrays of intersections of distance-regular graphs in which the neighborhoods of the vertices are pseudogeometric graphs for $pG_{s-3}(s,t)$ were found. In particular, the locally pseudo $pG_2(5,2)$-graph is a strongly regular graph with parameters $(117,36,15,9)$. The first main result of this paper is a theorem in which the possible orders and the structure of the subgraphs of fixed points of automorphisms of a strongly regular graph with parameters $(117,36,15,9)$ are found. This graph has a spectrum of $36^1,9^26,-3^90$. The order of clicks in $\Gamma$ does not exceed $1+36/3=13$, the order of the cocliques in $\Gamma$ does not exceed $117\cdot 3/39=9$. Further, from the obtained theorem, the following result is derived: if the group $\Gamma$ of automorphisms of a strongly regular graph with parameters $(117,36,15,9)$ acts transitively on the set of vertices, then the socle $T$ of the group $\Gamma$ is isomorphic to either $L_3(3)$ and $T_a\cong GL_2(3)$ is a subgroup of index $117$, or $T_a\cong GL_2(3)$ and $T_a\cong U_4(2).Z_2$ is a subgroup of index $117$.

Key words: strongly regular graph, symmetric graph, automorphism groups of graph.

DOI: https://doi.org/10.23671/VNC.2018.4.23386

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UDC: 519.17
MSC: 20D05

Citation: A. K. Gutnova, A. A. Makhnev, “On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$”, Vladikavkaz. Mat. Zh., 20:4 (2018), 43–49

Citation in format AMSBIB
\Bibitem{GutMak18} \by A.~K.~Gutnova, A.~A.~Makhnev \paper On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$ \jour Vladikavkaz. Mat. Zh. \yr 2018 \vol 20 \issue 4 \pages 43--49 \mathnet{http://mi.mathnet.ru/vmj675} \crossref{https://doi.org/10.23671/VNC.2018.4.23386}