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St. Petersburg Mathematical Journal

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

 
 

 

On edge-regular graphs with $ k\ge 3 b_1-3$


Authors: I. N. Belousov and A. A. Makhnev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 18 (2006), nomer 4.
Journal: St. Petersburg Math. J. 18 (2007), 517-538
MSC (2000): Primary 05C60
DOI: https://doi.org/10.1090/S1061-0022-07-00959-4
Published electronically: May 25, 2007
MathSciNet review: 2262582
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Abstract | References | Similar Articles | Additional Information

Abstract: An undirected graph on $ v$ vertices in which the degrees of all vertices are equal to $ k$ and each edge belongs to exactly $ \lambda$ triangles is said to be edge-regular with parameters $ (v,k,\lambda)$. It is proved that an edge-regular graph with parameters $ (v,k,\lambda)$ such that $ k\ge 3b_1-3$ either has diameter 2 and coincides with the graph $ P(2)$ on 20 vertices or with the graph $ M(19)$ on 19 vertices; or has at most $ 2k+4$ vertices; or has diameter at least 3 and is a trivalent graph without triangles, or the line graph of a quadrivalent graph without triangles, or a locally hexagonal graph; or has diameter 3 and satisfies $ \vert\Gamma_3(u)\vert\le 1$ for each vertex $ u$.


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Additional Information

I. N. Belousov
Affiliation: Institute of Mathematics and Mechanics, Ural Branch of RAS, 16 Kovalevskaya Street, Ekaterinburg, Russia 620219

A. A. Makhnev
Affiliation: Institute of Mathematics and Mechanics, Ural Branch of RAS, 16 Kovalevskaya Street, Ekaterinburg, Russia 620219
Email: makhnev@imm.uran.ru

DOI: https://doi.org/10.1090/S1061-0022-07-00959-4
Keywords: Undirected graph, edge-regular graph, locally hexagonal graph
Received by editor(s): June 27, 2005
Published electronically: May 25, 2007
Additional Notes: Supported by RFBR (grant no. 05-01-00046) and RFBR-NSFC (grant no. 05-01-39000)
Article copyright: © Copyright 2007 American Mathematical Society

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