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Lower bounds for the partitioning of graphs

Authors:

W. E. Donath,

A. J. HoffmanAuthors Info & Claims

IBM Journal of Research and Development, Volume 17, Issue 5

Pages 420 - 425

https://doi.org/10.1147/rd.175.0420

Published: 01 September 1973 Publication History

Abstract

Let a k-partition of a graph be a division of the vertices into k disjoint subsets containing m₁ ≥ m₂,..., ≥m_k vertices. Let E_c be the number of edges whose two vertices belong to different subsets. Let λ₁ ≥ λ₂, ..., ≥ λ_k, be the k largest eigenvalues of a matrix, which is the sum of the adjacency matrix of the graph plus any diagonal matrix U such that the suomf all the elements of the sum matrix is zero. Then E_c ≥ 1/2Σ_r=1^k-m_rλ_r.

A theorem is given that shows the effect of the maximum degree of any node being limited, and it is also shown that the right-hand side is a concave function of U.C omputational studies are madoef the ratio of upper bound to lower bound for the two-partition of a number of random graphs having up to 100 nodes.

References

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cover image IBM Journal of Research and Development

IBM Journal of Research and Development Volume 17, Issue 5

September 1973

85 pages

ISSN:0018-8646

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IBM Corp.

United States

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Published: 01 September 1973

Received: 20 March 1973

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Lower and upper bounds for long induced paths in 3-connected planar graphs

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