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Amply regular graphs and block designs

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Abstract

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ3(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    A. L. Gavrilyuk & A. A. Makhnev

Authors
  1. A. L. Gavrilyuk
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  2. A. A. Makhnev
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Original Russian Text Copyright © 2006 Gavrilyuk A. L. and Makhnev A. A.

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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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 753–768, July–August, 2006.

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Gavrilyuk, A.L., Makhnev, A.A. Amply regular graphs and block designs. Sib Math J 47, 621–633 (2006). https://doi.org/10.1007/s11202-006-0073-y

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  • DOI: https://doi.org/10.1007/s11202-006-0073-y

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