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Sibirskii Matematicheskii Zhurnal, 2006, Volume 47, Number 4, Pages 753–768
(Mi smj892)
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This article is cited in 4 scientific papers (total in 5 papers)
Amply regular graphs and block designs
A. L. Gavrilyuk, A. A. Makhnev Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
We study the amply regular diameter $d$ graphs $\Gamma$ 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 $\Gamma_3(a)$ 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, $t-(v,k,\lambda)$-design, strongly regular graph.
Received: 12.05.2004 Revised: 30.12.2005
Citation:
A. L. Gavrilyuk, A. A. Makhnev, “Amply regular graphs and block designs”, Sibirsk. Mat. Zh., 47:4 (2006), 753–768; Siberian Math. J., 47:4 (2006), 621–633
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https://www.mathnet.ru/eng/smj892 https://www.mathnet.ru/eng/smj/v47/i4/p753
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Abstract page: | 468 | Full-text PDF : | 133 | References: | 60 |
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