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Spectral Bounds for the k-Regular Induced Subgraph Problem

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Applied and Computational Matrix Analysis (MAT-TRIAD 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 192))

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Abstract

Many optimization problems on graphs are reduced to the determination of a subset of vertices of maximum cardinality which induces a k-regular subgraph . For example, a maximum independent set, a maximum induced matching and a maximum clique is a maximum cardinality 0-regular, 1-regular and \((\omega (G)-1)\)-regular induced subgraph , respectively, were \(\omega (G)\) denotes the clique number of the graph G. The determination of the order of a k-regular induced subgraph of highest order is in general an NP-hard problem . This paper is devoted to the study of spectral upper bounds on the order of these subgraphs which are determined in polynomial time and in many cases are good approximations of the respective optimal solutions. The introduced upper bounds are deduced based on adjacency, Laplacian and signless Laplacian spectra. Some analytical comparisons between them are presented. Finally, all of the studied upper bounds are tested and compared through several computational experiments.

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Acknowledgements

The authors would like to thank Willem Haemers for several insightful comments and suggestions on this work that have helped us to improve the content of this paper. This research was supported by the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), through the CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013. We are also indebted to the anonymous referee for her/his careful reading and suggestions which have improved the text.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Domingos Moreira Cardoso & Sofia J. Pinheiro

Authors
  1. Domingos Moreira Cardoso
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  2. Sofia J. Pinheiro
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Correspondence to Domingos Moreira Cardoso .

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Editors and Affiliations

  1. Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Natália Bebiano

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Cardoso, D.M., Pinheiro, S.J. (2017). Spectral Bounds for the k-Regular Induced Subgraph Problem. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_7

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