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Graphs and Combinatorics

, Volume 33, Issue 5, pp 1283–1295 | Cite as

Laplacian Distribution and Domination

  • Domingos M. Cardoso
  • David P. Jacobs
  • Vilmar TrevisanEmail author
Original Paper
  • 157 Downloads

Abstract

Let \(m_G(I)\) denote the number of Laplacian eigenvalues of a graph G in an interval I, and let \(\gamma (G)\) denote its domination number. We extend the recent result \(m_G[0,1) \le \gamma (G)\), and show that isolate-free graphs also satisfy \(\gamma (G) \le m_G[2,n]\). In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of \(\gamma (G)\), showing that \(\frac{\gamma (G)}{m_G[0,1)} \not \in O(\log n)\). However, \(\gamma (G) \le m_G[2, n] \le (c + 1) \gamma (G)\) for c-cyclic graphs, \(c \ge 1\). For trees T, \(\gamma (T) \le m_T[2, n] < 2 \gamma (G)\).

Keywords

Graph Laplacian eigenvalue Domination number 

Mathematics Subject Classification

05C50 05C69 

Notes

Acknowledgements

David P. Jacobs and Vilmar Trevisan were supported by CNPq Grant 400122/2014-6, Science without Borders, Brazil. Domingos M. Cardoso was partially 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. Vilmar Trevisan acknowledges the additional support of CNPq, Grants 409746/2016-9 and 303334/2016-9.

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Copyright information

© Springer Japan KK 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  2. 2.School of ComputingClemson UniversityClemsonUSA
  3. 3.Instituto de MatemáticaUFRGSPorto AlegreBrazil

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