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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2006, Volume 18, Issue 4, Pages 10–38 (Mi aa77)

Research Papers

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

I. N. Belousov, A. A. Makhnev

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

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 $|\Gamma_3(u)|\le 1$ for each vertex $u$.

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English version:
St. Petersburg Mathematical Journal, 2007, 18:4, 517–538

Bibliographic databases:

MSC: 05C60

Citation: I. N. Belousov, A. A. Makhnev, “On edge-regular graphs with $k\ge 3b_1-3$”, Algebra i Analiz, 18:4 (2006), 10–38; St. Petersburg Math. J., 18:4 (2007), 517–538

Citation in format AMSBIB
\Bibitem{BelMak06} \by I.~N.~Belousov, A.~A.~Makhnev \paper On edge-regular graphs with $k\ge 3b_1-3$ \jour Algebra i Analiz \yr 2006 \vol 18 \issue 4 \pages 10--38 \mathnet{http://mi.mathnet.ru/aa77} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2262582} \zmath{https://zbmath.org/?q=an:1157.05338} \elib{http://elibrary.ru/item.asp?id=9243971} \transl \jour St. Petersburg Math. J. \yr 2007 \vol 18 \issue 4 \pages 517--538 \crossref{https://doi.org/10.1090/S1061-0022-07-00959-4}