Skip to main content
SpringerLink
Log in

On the Eigenvalues Distribution in Threshold Graphs

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

Abstract

A threshold graph G of order n is defined by binary sequence of length n. In this paper, we consider the adjacent matrix of a connected threshold graph, and give the eigenvalues distribution in threshold graphs. Moreover, we pick out all the threshold graphs with distinct eigenvalues, and determine the HOMO–LUMO index of threshold graphs.

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

Fig. 1

Similar content being viewed by others

References

  1. Henderson, P.B., Zalcstein, Y.: A graph-theoretic characterization of the PV class of synchronizing primitives. SIAM J. Comput. 6, 88–108 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mahadev, V.N., Peled, U.N.: Threshold Graphs and Related Topics. Elsevier, Oxford (1995)

    MATH  Google Scholar 

  3. Banerjeea, A., Mehataria, R.: On the normalized spectrum of threshold graphs. Linear Algebra Appl. 530, 288–304 (2017)

    Article  MathSciNet  Google Scholar 

  4. Hammer, P.L., Kelmans, A.K.: Laplacian spectra and spanning trees of threshold graphs. Discret. Appl. Math. 65, 255–273 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lu, L., Huang, Q.X., Lou, Z.Z.: On the distance spectra of threshold graphs. Linear Algebra Appl. 553, 223–237 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sciriha, I., Farrugia, S.: On the spectrum of threshold graphs. ISRN Discret. Math. (2011)

  7. Bapat, R.B.: On the adjacency matrix of a threshold graph. Linear Algebra Appl. 439, 3008–3015 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalue location in threshold graphs. Linear Algebra Appl. 439, 2762–2773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jacobs, D.P., Trevisan, V., Tura, F.: Computing the characteristic polynomial of threshold Graphs. J. Graph Algorithm Appl. 18, 709–719 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalues and energy in threshold graphs. Linear Algebra Appl. 465, 412–425 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lazzarin, J., Márquez, O.F., Tura, F.: No threshold graphs are cospectral. Linear Algebra Appl. 560, 133–145 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harary, F., Schwenk, A.J.: Which graphs have integral spectra? In: Bari, R., Harary, F. (eds.) Graphs and Combinatorics. Lecture Notes in Mathematics, vol. 406, pp. 45–51. Springer, Berlin (1974)

    Chapter  Google Scholar 

  13. Mowshowitz, A.: Graphs, groups and matrices. In: Proceedings of 25th Summer Meeting Canadian Mathematical Congress, Congr. Numer. 4, Util. Math. Winnipeg, pp. 509–522 (1971)

  14. Lou, Z.Z., Huang, Q.X., Huang, X.Y.: Construction of graphs with distinct eigenvalues. Discret. Math. 340, 607–616 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tao, T., Vu, V.: Random matrices have simple spectrum. Combinatorica 37, 539–553 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fowler, P.W., Pisanski, T.: HOMO–LUMO maps for fullerenes. Acta Chim. Slov. 57, 513–517 (2010)

    MATH  Google Scholar 

  17. Fowler, P.W., Pisanski, T.: HOMO–LUMO maps for chemical graphs. MATCH Commun. Math. Comput. Chem. 64, 373–390 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Mohar, B.: Median eigenvalues and the HOMO–LUMO index of graphs. J. Combin. Theory, Ser. B 112, 78–92 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, K., Mohar, B.: Large regular bipartite graphs with median eigenvalue 1. Linear Algebra Appl. 449, 68–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, X.L., Li, Y.Y., Shi, Y.T., Gutman, I.: Note on the HOMOCLUMO index of graphs. MATCH Commun. Math. Comput. Chem. 70, 85–96 (2013)

    MathSciNet  Google Scholar 

  21. Mohar, B.: Median eigenvalues of bipartite planar graphs. MATCH Commun. Math. Comput. Chem. 70, 79–84 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Mohar, B., Tayfeh-Rezaie, B.: Median eigenvalues of bipartite graphs. J. Algebr. Combin. 41, 899–909 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mohar, B.: Median eigenvalues of bipartite subcubic graphs. Combin. Probab. Comput. 25, 768–790 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ye, D., Yang, Y.J., Mandal, B., Klein, D.J.: Graph invertibility and median eigenvalues. Linear Algebra Appl. 513, 304–323 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Godsil, C.D., Royle, G.: Algebraic Graph Theory. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  26. Aguilar, C.O., Lee, J., Piato, E., Schweitzer, B.J.: Spectral characterizations of anti-regular graphs. Linear Algebra Appl. 557, 84–104 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their valuable suggestions which lead to a great improvement in the original paper. The research is supported by National Natural Science Foundation of China (nos. 11701492, 11671344, 11461054).

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Xinjiang University, Ürümqi, China

    Zhenzhen Lou & Qiongxiang Huang

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, China

    Jianfeng Wang

Authors
  1. Zhenzhen Lou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianfeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiongxiang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianfeng Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lou, Z., Wang, J. & Huang, Q. On the Eigenvalues Distribution in Threshold Graphs. Graphs and Combinatorics 35, 867–880 (2019). https://doi.org/10.1007/s00373-019-02042-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-019-02042-1

Keywords

Search

Navigation