SpringerLink
Log in

Graphs with Three Distinct α-Eigenvalues

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

In this paper, we investigate properties of graphs with three distinct α-eigenvalues of the matrix Aα. In particular, we show for some α the connected graph G = Kn − e,G = K1 ∨ (nKn) and some cones over strongly regular graph admit three distinct α-eigenvalues.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayoobi, F.M., Omidi, G.R., Tayfeh-Rezaie, B.: A note on graphs whose signless Laplacian has three distinct eigenvalues. Linear Multilinear Algebra 59(6), 701–706 (2011)

    Article  MathSciNet  Google Scholar 

  2. Bridges, W.G., Mena, R.A.: Multiplicative cones—a family of three eigenvalue graphs. Aequ. Math. 22(2–3), 208–214 (1981)

    Article  MathSciNet  Google Scholar 

  3. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)

    Book  Google Scholar 

  4. Cui, S.-Y., Tian, G.-X.: The spectrum and the signless Laplacian spectrum of coronae. Linear Algebra Appl. 437(7), 1692–1703 (2012)

    Article  MathSciNet  Google Scholar 

  5. Haemers, W.H., Omidi, G.R.: Universal adjacency matrices with two eigenvalues. Linear Algebra Appl. 435(10), 2520–2529 (2011)

    Article  MathSciNet  Google Scholar 

  6. Hahn, G., Sabidussi, G.: Graph Symmetry–Algebraic Methods and Applications. Kluwer Acdemic Publishers, London (1996)

    Google Scholar 

  7. Lin, H., Liu, X.G., Xue, J.: Graphs determined by their A α spectra. ar**v:1709.00792v1

  8. Muzychuk, M., Klin, M.: On graphs with three eigenvalues. Discret. Math. 189(1–3), 191–207 (1998)

    Article  MathSciNet  Google Scholar 

  9. Nikiforov, V.: Merging the A- and Q-spectral theories. Appl. Anal. Discrete Math. 11(1), 81–107 (2017)

    Article  MathSciNet  Google Scholar 

  10. van Dam, E.R.: Nonregular graphs with three eigenvalues. J. Combin. Theory Ser. B 73(2), 101–118 (1998)

    Article  MathSciNet  Google Scholar 

  11. van Dam, E.R., Haemers, W.H.: Graphs with constant μ and \(\overline {\mu }\). Discret. Math. 182(1–3), 293–307 (1998)

    Google Scholar 

  12. van Dam, E.R., Haemers, W.H.: Which graphs are determined by their spectrum? Linear Algebra Appl. 373, 241–272 (2003)

    Article  MathSciNet  Google Scholar 

  13. van Dama, E.R., Omidi, G.R.: Graphs whose normalized Laplacian has three eigenvalues. Linear Algebra Appl. 435(10), 2560–2569 (2011)

    Article  MathSciNet  Google Scholar 

  14. Wang, Y., Fan, Y., Tan, Y.: On graphs with three distinct Laplacian eigenvalues. Appl. Math. J. Chin. Univ. Ser. B 22(4), 478–484 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for many helpful comments and suggestions to the earlier version of this paper.

Funding

This work is supported by the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001)), the National Natural Science Foundation of China (No. 11531001).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, MOE-LSC, SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Muhammad Ateeq Tahir & **ao-Dong Zhang

Authors
  1. Muhammad Ateeq Tahir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. **ao-Dong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **ao-Dong Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tahir, M.A., Zhang, XD. Graphs with Three Distinct α-Eigenvalues. Acta Math Vietnam 43, 649–659 (2018). https://doi.org/10.1007/s40306-018-0275-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-018-0275-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation