Skip to main content
SpringerLink
Log in

Some Relations Between the Eigenvalues of Adjacency, Laplacian and Signless Laplacian Matrix of a Graph

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G) = D(G) − A(G) and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). Let λ 1(G), μ 1(G) and q 1(G) denote the largest eigenvalues of A(G), L(G) and Q(G), respectively. In this paper, we give upper and lower bounds on q 1(G) − μ 1(G), q 1(G) − λ 1(G) and μ 1(G) − λ 1(G), moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, we give a sharp lower bound on q 1(G) + q 2(G).

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. Aouchiche M., Hansen P., Lucas C.: On the extremal values of the second largest Q-eigenvalue. Linear Algebra Appl. 435, 2591–2606 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen Y.: Properties of spectra of graphs and line graphs. Appl. Math. J. Ser. B 3, 371–376 (2002)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Cvetković D., Simić S.K.: Towards a spectral theory of graphs based on signless Laplacian, I. Publ. Inst. Math. (Beograd) (N.S.) 85(99), 19–33 (2009)

    Article  MathSciNet  Google Scholar 

  5. Cvetković D., Simić S.K.: Towards a spectral theory of graphs based on signless Laplacian, II. Linear Algebra Appl. 432, 2257–2272 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cvetković D., Simić S.K.: Towards a spectral theory of graphs based on signless Laplacian, III. Appl. Anal. Discret. Math. 4, 156–166 (2010)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Cvetković D., Rowlinson P., Simić S.K.: Eigenvalue bounds for the signless Laplacian. Publ. Inst. Math. (Beograd) (N.S.) 81(95), 11–27 (2007)

    Article  MathSciNet  Google Scholar 

  9. Das K.C.: Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs. Linear Algebra Appl. 435, 2420–2424 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Das K.C.: On conjectures involving second largest signless Laplacian eigenvalue of graphs. Linear Algebra Appl. 432, 3018–3029 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Grone R., Meris R., Sunder V.S.: The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11, 218–238 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grone R., Merris R.: The Laplacian spectrum of a graph II. SIAM J. Discret. Math. 7(2), 221–229 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nikiforov V.: Chromatic number and spectral radius. Linear Algebra Appl. 426, 810–814 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Schott, J.R.: Matrix analysis for statistics, 2nd edn. In: Wiley Series in Probability and Statistics. Wiley, Hoboken (2005)

  16. Smith J.H.: Some Properties of the Spectrum of a Graph, Combinatorial Structures and Their Applications, pp. 403–406. Gordon and Breach, New York (1970)

    Google Scholar 

  17. So, W.: Commutativity and spectra of Hermitian matrices. Linear Algebra Appl. 212–213, 121–129 (1994)

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Huiqiu Lin

  2. Department of Mathematics, East China Normal University, Shanghai, 200062, China

    Yuan Hong & Jinlong Shu

Authors
  1. Huiqiu Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinlong Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huiqiu Lin.

Additional information

This was supported by the National Natural Science Foundation of China (No. 11071078, 11075057, 11371137).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, H., Hong, Y. & Shu, J. Some Relations Between the Eigenvalues of Adjacency, Laplacian and Signless Laplacian Matrix of a Graph. Graphs and Combinatorics 31, 669–677 (2015). https://doi.org/10.1007/s00373-013-1398-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-013-1398-5

Keywords

Mathematics Subject Classification

Search

Navigation