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On almost good triples of vertices in edge regular graphs

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Abstract

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b 1 = k−λ−1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k − 4b 1 + 3 (and µ(u, w) + µ(u, z) = 2k − 4b 1 + 4). If k = 3b 1 + γ with γ ≥ −2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b 1 = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    V. I. Belousova & A. A. Makhnev

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  1. V. I. Belousova
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  2. A. A. Makhnev
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Correspondence to A. A. Makhnev.

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Original Russian Text Copyright © 2011 Belousova V. I. and Makhnev A. A.

The authors were supported by the Russia-Slovenia Project for 2010–2011, the Program of the Division of Mathematical Sciences of the Russian Academy of Sciences (Grant 09-T-1-1004), and the Joint Program of the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences and the Sobolev Institute of Mathematics of the Siberian Division (Grant 09-C-1-1007) with the National Academy of Sciences of the Republic of Belarus (Grant 09-C-1-1009).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 4, pp. 745–753, July–August, 2011.

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Belousova, V.I., Makhnev, A.A. On almost good triples of vertices in edge regular graphs. Sib Math J 52, 585–592 (2011). https://doi.org/10.1134/S0037446611040033

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