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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

On locally $ GQ(s,t)$ graphs with strongly regular $ \mu$-subgraphs


Authors: V. I. Kazarina and A. A. Makhnev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 17 (2005), nomer 3.
Journal: St. Petersburg Math. J. 17 (2006), 443-452
MSC (2000): Primary 05C75
DOI: https://doi.org/10.1090/S1061-0022-06-00913-7
Published electronically: March 9, 2006
MathSciNet review: 2167845
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Abstract: The connected locally $ GQ(s,t)$ graphs are studied in which every $ \mu$-subgraph is a known strongly regular graph (i.e., $ K_{m,m}$ for a positive integer $ m$, the Moore graph with parameters $ (k^2+1,k,0,1)$, $ k=2,3$, or $ 7$, the Clebsch graph, the Gewirtz graph, the Higman-Sims graph, or the second neighborhood (with parameters $ (77,16,0,4))$ of a vertex in the Higman-Sims graph). It is proved that if $ \Gamma$ is a strongly regular locally $ GQ(s,t)$ graph in which every $ \mu$-subgraph is isomorphic to a known strongly regular graph $ \Delta$, then one of the following statements is true: $ (1)$ $ \Delta=K_{t+1,t+1}$ and either $ s=1$ and $ \Gamma=K_{3\times (t+1)}$, or $ s=4,t=1$, and $ \Gamma$ is a quotient of the Johnson graph $ \overline{J}(10,5)$, or $ s=t=1,2,3,8,13$; $ (2)$ $ \Delta$ is a Petersen graph and $ \Gamma$ is a unique locally $ GQ(2,2)$ graph with parameters $ (28,15,6,10)$; $ (3)$ $ \Delta$ is the Gewirtz graph and $ \Gamma$ is the McLaughlin graph.


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Additional Information

V. I. Kazarina
Affiliation: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

A. A. Makhnev
Affiliation: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
Email: makhnev@imm.uran.ru

DOI: https://doi.org/10.1090/S1061-0022-06-00913-7
Keywords: Strongly regular graphs, locally $\mathcal{F}$ graphs, geometry of rank $2$
Received by editor(s): January 10, 2004
Published electronically: March 9, 2006
Additional Notes: This work was supported by RFBR (grant 02–01–00772).
Article copyright: © Copyright 2006 American Mathematical Society