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 Chebyshevskii Sb., 2016, Volume 17, Issue 3, Pages 178–185 (Mi cheb505)

On automorphisms of strongly regular graph with the parametrs $(1276,50,0,2)$

V. V. Nosov

Orenburg State University

Abstract: Let $\Gamma$ be a strongly regular graph with parameters $(v,k,0,2)$. Then $k=u^2+1$, $v=(u^4+3u^2+4)/2$ and $u \equiv 1, 2, 3(mod 4)$. If $u=1$, then $\Gamma$ has parametrs $(4,2,0,2)$ — tetragonal graph. If $u=2$, then $\Gamma$ has parametrs $(15,5,0,2)$ — Clebsch graph. If $u=3$, then $\Gamma$ has parametrs $(56,10,0,2)$ — Gewirtz graph. If $u=5$ then hypothetical strongly regular graph$\Gamma$ has parametrs $(352,26,0,2)$ [4]. If $u=5$ then hypothetical strongly regular graph$\Gamma$ has parametrs $(704,37,0,2)$ [5]. Let $u=7$, then $\Gamma$ has parametrs $(1276,50,0,2)$. Let $G$ be the automorphism group of a hypothetical strongly regular graph with parameters $(1276, 50, 0, 2)$. Possible orders are found and the structure of fixed-point subgraphs is determined for elements of prime order in $G$. With the use of theory of characters of finite groups we find the possible orders and the structures of subgraphs of the fixed points of automorphisms of the graph with parameters $(1276,50,0,2)$. It proved that if the graph with parametrs $(1276,50,0,2)$ exist, its automorphism group divides $2^l\cdot 3\cdot 5^m\cdot 7\cdot 11\cdot 29$. In particulary, $G$ — solvable group.
Bibliography: 17 titles.

Keywords: strongly regular graph, prime order automorphisms of strongly regular graph, fixed-point subgraphs.

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UDC: 519.17+512.54
Accepted:13.09.2016

Citation: V. V. Nosov, “On automorphisms of strongly regular graph with the parametrs $(1276,50,0,2)$”, Chebyshevskii Sb., 17:3 (2016), 178–185

Citation in format AMSBIB
\Bibitem{Nos16} \by V.~V.~Nosov \paper On automorphisms of strongly regular graph with the parametrs $(1276,50,0,2)$ \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 3 \pages 178--185 \mathnet{http://mi.mathnet.ru/cheb505} \elib{https://elibrary.ru/item.asp?id=27452090} 

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This publication is cited in the following articles:
1. V. G. Chirskii, “Periodicheskie i neperiodicheskie konechnye posledovatelnosti”, Chebyshevskii sb., 18:2 (2017), 275–278
2. E. S. Krupitsyn, “Otsenka mnogochlena ot globalno transtsendentnogo poliadicheskogo chisla”, Chebyshevskii sb., 18:4 (2017), 256–260
3. A. Kh. Munos Vaskes, “O $q$-ichnykh periodicheskikh posledovatelnostyakh”, Trudy mezhdunarodnoi konferentsii «Klassicheskaya i sovremennaya geometriya», posvyaschennoi 100-letiyu so dnya rozhdeniya professora Vyacheslava Timofeevicha Bazyleva. Moskva, 22–25 aprelya 2019 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 179, VINITI RAN, M., 2020, 34–36
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