Advertisement

Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 649–659 | Cite as

Graphs with Three Distinct α-Eigenvalues

  • Muhammad Ateeq Tahir
  • Xiao-Dong Zhang
Article
  • 53 Downloads

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 = Kne,G = K1 ∨ (nKn) and some cones over strongly regular graph admit three distinct α-eigenvalues.

Keywords

Aα matrix α-eigenvalues Join of graphs Strongly regular graph 

Mathematics Subject Classification (2010)

05C50 05C75 

Notes

Acknowledgements

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

Funding information

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).

References

  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bridges, W.G., Mena, R.A.: Multiplicative cones—a family of three eigenvalue graphs. Aequ. Math. 22(2–3), 208–214 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)CrossRefGoogle Scholar
  4. 4.
    Cui, S.-Y., Tian, G.-X.: The spectrum and the signless Laplacian spectrum of coronae. Linear Algebra Appl. 437(7), 1692–1703 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Haemers, W.H., Omidi, G.R.: Universal adjacency matrices with two eigenvalues. Linear Algebra Appl. 435(10), 2520–2529 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hahn, G., Sabidussi, G.: Graph Symmetry–Algebraic Methods and Applications. Kluwer Acdemic Publishers, London (1996)Google Scholar
  7. 7.
    Lin, H., Liu, X.G., Xue, J.: Graphs determined by their A α spectra. arXiv:1709.00792v1
  8. 8.
    Muzychuk, M., Klin, M.: On graphs with three eigenvalues. Discret. Math. 189(1–3), 191–207 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nikiforov, V.: Merging the A- and Q-spectral theories. Appl. Anal. Discrete Math. 11(1), 81–107 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    van Dam, E.R.: Nonregular graphs with three eigenvalues. J. Combin. Theory Ser. B 73(2), 101–118 (1998)MathSciNetCrossRefGoogle Scholar
  11. 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. 12.
    van Dam, E.R., Haemers, W.H.: Which graphs are determined by their spectrum? Linear Algebra Appl. 373, 241–272 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    van Dama, E.R., Omidi, G.R.: Graphs whose normalized Laplacian has three eigenvalues. Linear Algebra Appl. 435(10), 2560–2569 (2011)MathSciNetCrossRefGoogle Scholar
  14. 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences, MOE-LSCSHL-MAC, Shanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

Personalised recommendations