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Sibirsk. Mat. Zh., 2007, Volume 48, Number 4, Pages 817–832 (Mi smj1747)  

This article is cited in 4 scientific papers (total in 5 papers)

A new estimate for the vertex number of an edge-regular graph

A. A. Makhnev, D. V. Paduchikh

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

Abstract: Given a connected edge-regular graph $\Gamma$ with parameters $(v,k,\lambda)$ and $b_1=k-\lambda-1$, we prove that in the case $k\geqslant3b_1-2$ either $|\Gamma_2(u)|(k-2b_1+2)<kb_1$ for every vertex $u$ or $\Gamma$ is a polygon, the edge graph of a trivalent graph without triangles that has diameter greater than 2, the icosahedral graph, the complete multipartite graph $K_{r\times2}$, the $3\times3$-grid, the triangular graph $T(m)$ with $m\leqslant7$, the Clebsch graph, or the Schläfli graph.

Keywords: edge-regular graph, characterization by parameters.

Full text: PDF file (447 kB)
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English version:
Siberian Mathematical Journal, 2007, 48:4, 653–665

Bibliographic databases:

Received: 22.11.2005

Citation: A. A. Makhnev, D. V. Paduchikh, “A new estimate for the vertex number of an edge-regular graph”, Sibirsk. Mat. Zh., 48:4 (2007), 817–832; Siberian Math. J., 48:4 (2007), 653–665

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Makhnev, N. V. Chuksina, “O khoroshikh parakh vershin v reberno regulyarnykh grafakh s $k=3b_1-1$”, Tr. IMM UrO RAN, 14, no. 4, 2008, 119–134  mathnet  elib
    2. Konstantin S. Efimov, Aleksandr A. Makhnev, “Vpolne regulyarnye grafy s $b_1=6$”, Zhurn. SFU. Ser. Matem. i fiz., 2:1 (2009), 63–77  mathnet  elib
    3. V. I. Belousova, A. A. Makhnev, “On almost good triples of vertices in edge regular graphs”, Siberian Math. J., 52:4 (2011), 585–592  mathnet  crossref  mathscinet  isi
    4. “Makhnev Aleksandr Alekseevich (on his 60th birthday)”, Proc. Steklov Inst. Math. (Suppl.), 285, suppl. 1 (2014), 1–11  mathnet  crossref  mathscinet
    5. A. V. Mityanina, “O $K_{1,3}$-svobodnykh grafakh Deza diametra bolshe dvukh”, Tr. IMM UrO RAN, 20, no. 2, 2014, 238–241  mathnet  mathscinet  elib
  • Сибирский математический журнал Siberian Mathematical Journal
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