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 Algebra Logika, 2005, Volume 44, Number 3, Pages 335–354 (Mi al116)  Automorphisms of Strongly Regular Krein Graphs without Triangles

A. A. Makhnev, V. V. Nosov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: A strongly regular graph is called a Krein graph if, in one of the Krein conditions, an equality obtains for it. A strongly regular Krein graph $Kre(r)$ without triangles has parameters $((r^2+3r)^2,r^3+3r^2+r,0,r^2+r)$. It is known that $Kre(1)$ is a Klebsh graph, $Kre(2)$ is a Higman –Sims graph, and that a graph of type $Kre(3)$ does not exist. Let $G$ be the automorphism group of a hypothetical graph $\Gamma=Kre(5)$, $g$ be an element of odd prime order $p$ in $G$, and $\Omega=\operatorname{Fix}(g)$. It is proved that either $\Omega$ is the empty graph and $p=5$, or $\Omega$ is a one-vertex graph and $p=41$, or $\Omega$ is a $2$-clique and $p=17$, or $\Omega$ is the complete bipartite graph $K_{8,8}$, from which the maximal matching is removed, and $p=3$.

Keywords: automorphism, Krein graph, Klebsh graph, Higman – Sims graph, $n$-clique, $n$-coclique Full text: PDF file (230 kB) References: PDF file   HTML file

English version:
Algebra and Logic, 2005, 44:3, 185–196 Bibliographic databases:   UDC: 519.14
Revised: 12.01.2005

Citation: A. A. Makhnev, V. V. Nosov, “Automorphisms of Strongly Regular Krein Graphs without Triangles”, Algebra Logika, 44:3 (2005), 335–354; Algebra and Logic, 44:3 (2005), 185–196 Citation in format AMSBIB
\Bibitem{MakNos05} \by A.~A.~Makhnev, V.~V.~Nosov \paper Automorphisms of Strongly Regular Krein Graphs without Triangles \jour Algebra Logika \yr 2005 \vol 44 \issue 3 \pages 335--354 \mathnet{http://mi.mathnet.ru/al116} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2170690} \zmath{https://zbmath.org/?q=an:1103.05090} \transl \jour Algebra and Logic \yr 2005 \vol 44 \issue 3 \pages 185--196 \crossref{https://doi.org/10.1007/s10469-005-0019-7} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-22344432285} 

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This publication is cited in the following articles:
1. A. L. Gavrilyuk, “Ob avtomorfizmakh silno regulyarnogo grafa s parametrami $(784,116,0,20)$”, Sib. elektron. matem. izv., 5 (2008), 80–87  2. A. A. Makhnev, “On automorphisms of distance-regular graphs”, J. Math. Sci., 166:6 (2010), 733–742    3. “Makhnev Aleksandr Alekseevich (on his 60th birthday)”, Proc. Steklov Inst. Math. (Suppl.), 285, suppl. 1 (2014), 1–11   •  Number of views: This page: 252 Full text: 66 References: 25 First page: 1 Contact us: math-net2019_06 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019