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Siberian Mathematical Journal

, Volume 47, Issue 4, pp 621–633 | Cite as

Amply regular graphs and block designs

  • A. L. Gavrilyuk
  • A. A. Makhnev
Article

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.

Keywords

amply regular graph, t-(v, k, λ)-design, strongly regular graph 

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References

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. L. Gavrilyuk
    • 1
  • A. A. Makhnev
    • 1
  1. 1.Institute of Mathematics and MechanicsEkaterinburgRussia

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