Graphs and Combinatorics

, Volume 35, Issue 4, pp 867–880 | Cite as

On the Eigenvalues Distribution in Threshold Graphs

  • Zhenzhen Lou
  • Jianfeng WangEmail author
  • Qiongxiang Huang
Original Paper


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.


Adjacency matrix Threshold graph Median eigenvalues HOMO–LUMO Inertia 



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).


  1. 1.
    Henderson, P.B., Zalcstein, Y.: A graph-theoretic characterization of the PV class of synchronizing primitives. SIAM J. Comput. 6, 88–108 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mahadev, V.N., Peled, U.N.: Threshold Graphs and Related Topics. Elsevier, Oxford (1995)zbMATHGoogle Scholar
  3. 3.
    Banerjeea, A., Mehataria, R.: On the normalized spectrum of threshold graphs. Linear Algebra Appl. 530, 288–304 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hammer, P.L., Kelmans, A.K.: Laplacian spectra and spanning trees of threshold graphs. Discret. Appl. Math. 65, 255–273 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lu, L., Huang, Q.X., Lou, Z.Z.: On the distance spectra of threshold graphs. Linear Algebra Appl. 553, 223–237 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sciriha, I., Farrugia, S.: On the spectrum of threshold graphs. ISRN Discret. Math. (2011)Google Scholar
  7. 7.
    Bapat, R.B.: On the adjacency matrix of a threshold graph. Linear Algebra Appl. 439, 3008–3015 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalue location in threshold graphs. Linear Algebra Appl. 439, 2762–2773 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jacobs, D.P., Trevisan, V., Tura, F.: Computing the characteristic polynomial of threshold Graphs. J. Graph Algorithm Appl. 18, 709–719 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalues and energy in threshold graphs. Linear Algebra Appl. 465, 412–425 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lazzarin, J., Márquez, O.F., Tura, F.: No threshold graphs are cospectral. Linear Algebra Appl. 560, 133–145 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Lou, Z.Z., Huang, Q.X., Huang, X.Y.: Construction of graphs with distinct eigenvalues. Discret. Math. 340, 607–616 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tao, T., Vu, V.: Random matrices have simple spectrum. Combinatorica 37, 539–553 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fowler, P.W., Pisanski, T.: HOMO–LUMO maps for fullerenes. Acta Chim. Slov. 57, 513–517 (2010)zbMATHGoogle Scholar
  17. 17.
    Fowler, P.W., Pisanski, T.: HOMO–LUMO maps for chemical graphs. MATCH Commun. Math. Comput. Chem. 64, 373–390 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mohar, B.: Median eigenvalues and the HOMO–LUMO index of graphs. J. Combin. Theory, Ser. B 112, 78–92 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guo, K., Mohar, B.: Large regular bipartite graphs with median eigenvalue 1. Linear Algebra Appl. 449, 68–75 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetGoogle Scholar
  21. 21.
    Mohar, B.: Median eigenvalues of bipartite planar graphs. MATCH Commun. Math. Comput. Chem. 70, 79–84 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mohar, B., Tayfeh-Rezaie, B.: Median eigenvalues of bipartite graphs. J. Algebr. Combin. 41, 899–909 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mohar, B.: Median eigenvalues of bipartite subcubic graphs. Combin. Probab. Comput. 25, 768–790 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ye, D., Yang, Y.J., Mandal, B., Klein, D.J.: Graph invertibility and median eigenvalues. Linear Algebra Appl. 513, 304–323 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Godsil, C.D., Royle, G.: Algebraic Graph Theory. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Aguilar, C.O., Lee, J., Piato, E., Schweitzer, B.J.: Spectral characterizations of anti-regular graphs. Linear Algebra Appl. 557, 84–104 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiChina
  2. 2.School of Mathematics and StatisticsShandong University of TechnologyZiboChina

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