Siberian Mathematical Journal

, Volume 47, Issue 4, pp 621–633

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

1. 1.
Brouwer A. E., Cohen A. M., and Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin; Heidelberg; New York (1989).Google Scholar
2. 2.
Cameron P. J. and van Lint J. H., Graphs, Codes and Designs, Cambridge Univ. Press, Cambridge (1980) (London Math. Soc. Lecture Notes; 43).Google Scholar
3. 3.
Makhnev A. A., “On extensions of the partial geometries containing small μ-subgraphs, ” Diskret. Anal. Issled. Oper., 3, No. 3, 71–83 (1996).
4. 4.
Bous R. C. and Dowling T. A., “A generalization of Moore graphs of diameter 2,” J. Combin. Theory Ser. B, 11, No. 3, 213–226 (1971).

Authors and Affiliations

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

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