Skip to main content
SpringerLink
Log in

A Sharp Lower Bound on the Least Signless Laplacian Eigenvalue of a Graph

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

Let G be a simple graph with n vertices and m edges, and let \(q_{n}(G)\) be the least signless Laplacian eigenvalue of G. Recently, Guo, Chen and Yu proved that if \(n\ge 6\), then

$$\begin{aligned} q_{n}(G)\ge \frac{2m}{n-2}-n+1, \end{aligned}$$

which confirms a conjecture proposed by de Lima, Oliveira, de Abreu and Nikiforov. In this note, we present a new proof for the above inequality, where we also completely characterize the case when the equality holds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, W.N., Morley, T.D.: Eigenvalues of the Laplacian of a graph. Linear Multilinear Algebra 18, 141–145 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cardoso, D.M., Cvetković, D., Rowlinson, P., Simić, S.K.: A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph. Linear Algebra Appl. 429, 2770–2780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cvetković, D., Doob, M., Gutman, I., Torgas̆ev, A.: Recent Results in the Theory of Graph Spectra. Annals of Discrete Mathematics, vol. 36. North-Holland, Amsterdam (1988)

    Google Scholar 

  4. Cvetković, D., Rowlinson, P., Simić, S.K.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423, 155–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cvetković, D.: Spectral theory of graphs based on the signless Laplacian. Research report. http://www.mi.sanu.ac.rs/projects/signless_L_reportDec06 (2010)

  6. Cvetković, D., Rowlinson, P., Simić, S.K.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  7. Das, K.C.: The Laplacian spectrum of a graph. Comput. Math. Appl. 48, 715–724 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Lima, L.S., Oliveira, C.S., de Abreu, N.M.M., Nikiforov, V.: The smallest eigenvalue of the signless Laplacian. Linear Algebra Appl. 435, 2570–2584 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Desai, M., Rao, V.: A characterization of the smallest eigenvalue of a graph. J. Graph Theory 18, 181–194 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feng, L., Yu, G.: On three conjectures involving the signless Laplacian spectral radius of graphs. Publ. Inst. Math. (Beograd) 85(99), 35–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, J.: On the Laplacian spectral radius of a tree. Linear Algebra Appl. 368, 379–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, S., Chen, Y., Yu, G.: A lower bound of the least signless Laplacian eigenvalue of a graph. Linear Algebra Appl. 448, 217–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197&198, 143–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Oliveira, C.S., de Lima, L.S., de Abreu, N.M.M., Kirkland, S.: Bounds on the Q-spread of a graph. Linear Algebra Appl. 432, 2342–2351 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tan, X., Liu, B.: On the spectrum of the quasi-Laplacian matrix of a graph. Australas. J. Combin. 34, 49–55 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their helpful comments and suggestions toward improving the original version of the paper. This work was supported in part by National Natural Science Foundation of China (Nos. 11501133, 11571101), China Postdoctoral Science Foundation (No. 2015M572252) and Guangxi Natural Science Foundation (Nos. 2014GXNSFBA118008, 2016GXNSFAA380293).

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, People’s Republic of China

    Xiaodan Chen

  2. Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications, Nanning, 530004, People’s Republic of China

    Xiaodan Chen

  3. Department of Mathematics, Hunan Normal University, Changsha, 410081, People’s Republic of China

    Yaoping Hou

Authors
  1. Xiaodan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaoping Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaodan Chen.

Additional information

Communicated by Dr. Sanming Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Hou, Y. A Sharp Lower Bound on the Least Signless Laplacian Eigenvalue of a Graph. Bull. Malays. Math. Sci. Soc. 41, 2011–2018 (2018). https://doi.org/10.1007/s40840-016-0440-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-016-0440-1

Keywords

Search

Navigation