Advertisement

On the Distance and Distance Signless Laplacian Spectral Radii of Tricyclic Graphs

  • Zhongxun ZhuEmail author
  • Xin Zou
  • Yunchao Hong
Article
  • 7 Downloads

Abstract

In this paper, we first obtain the second lower bound on Wiener index for tricyclic graphs. As applications, those graphs with the first four minimum distance (resp. distance signless Laplacian) spectral radius among tricyclic graphs are characterized.

Keywords

Distance matrix Distance signless Laplacian Spectral radius Wiener index Tricyclic graph 

Mathematics Subject Classification

05C50 15A18 

Notes

Acknowledgements

The authors would like to express sincere gratitude to the editor and the reviewers for helpful comments in improving the quality of the original manuscript.

References

  1. 1.
    Bollobás, B.: Morden Graph Theory. Springer, Berlin (1998)Google Scholar
  2. 2.
    Aouchiche, M., Hansen, P.: Two Laplacians for the distance matrix of a graph. Linear Algebra Appl. 439, 21–33 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Graham, R.L., Pollack, H.O.: On the addressing problem for loop switching. Bell Syst. Tech. J. 50, 2495–2519 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Graham, R.L., Lovász, L.: Distance matrix polynomials of trees. Adv. Math. 29, 60–88 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Stevanović, D., IIić, A.: Distance spectral radius of trees with fixed maximum degree. Electron. J. Linear Algebra 20, 168–179 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Yu, G., Wu, Y., Zhang, Y., Shu, J.: Some graft transformations and its application on a distance spectrum. Discret. Math. 311, 2117–2123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Paul, S.: On the maximal distance spectral radius in a class of bicyclic graphs. Discret. Math. Algorithms Appl. 4, 1250061 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bose, S.S., Nath, M., Paul, S.: On the maximal distance spectral radius of graphs without a pendent vertex. Linear Algebra Appl. 438, 4260–4278 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nath, M., Paul, S.: On the distance spectral radius of bipartite graphs. Linear Algebra Appl. 436, 1285–1296 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nath, M., Paul, S.: On the distance spectral radius of trees. Linear Multilinear Algebra 61, 847–855 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhou, B., Ilić, A.: On distance spectral radius and distance energy of graphs. MATCH Commun. Math. Comput. Chem. 64, 261–280 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Xing, R., Zhou, B., Li, J.: On the distance signless Laplacian spectral radius of graphs. Linear Multilinear Algebra 62, 1377–1387 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xing, R., Zhou, B.: On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra Appl. 439, 3955–3963 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Aouchiche, M., Hansen, P.: On the distance signless Laplacian of a graph. Linear Multilinear Algebra 64(6), 1113–1123 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Das, K.C.: Proof of conjectures on the distance signless Laplacian eigenvalues of graphs. Linear Algebra Appl. 467, 100–115 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin, H., Lu, X.: Bounds on the distance signless Laplacian spectral radius in terms of clique number. Linear Multilinear Algebra 63(9), 1750–1759 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, S., Li, X., Zhu, Z.: On tricyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 59, 397–419 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Wang, D., Tan, S., Zhu, L.: On the lower and upper bounds for different indices of tricyclic graphs. J. Appl. Math. Comput. 51, 1–11 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and StatisticsSouth Central University for NationalitiesWuhanPeople’s Republic of China

Personalised recommendations