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

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Abstract

Given a connected edge-regular graph Γ with parameters (v, k, λ) and b 1 = k − λ − 1, we prove that in the case k ≥ 3b 1 −2 either |Γ2(u)|(k−2b 1 + 2) < kb 1 for every vertex u or Γ 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×2, the 3 × 3-grid, the triangular graph T(m) with m ≤ 7, the Clebsch graph, or the Schläfli graph.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    A. A. Makhnev & D. V. Paduchikh

Authors
  1. A. A. Makhnev
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  2. D. V. Paduchikh
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Additional information

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 4, pp. 817–832, July–August, 2007.

Original Russian Text Copyright © 2007 Makhnev A. A. and Paduchikh D. V.

The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00046) and RFFI-GFEN (Grant 05-01-39000).

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Makhnev, A.A., Paduchikh, D.V. A new estimate for the vertex number of an edge-regular graph. Sib Math J 48, 653–665 (2007). https://doi.org/10.1007/s11202-007-0067-4

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  • DOI: https://doi.org/10.1007/s11202-007-0067-4

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