Skip to main content

Advertisement

SpringerLink
Log in

On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

Let \(G=(V,E)\) be a simple graph of order n with m edges and Laplacian eigenvalues \(\mu _1\ge \mu _2\ge \cdots \ge \mu _{n-1}\ge \mu _n=0\). The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as

$$\begin{aligned} \mathrm{Kf}(G)=n\sum _{k=1}^{n-1}\frac{1}{\mu _k}~~ \text{ and } ~~\mathrm{LEL}(G)=\sum _{k=1}^{n-1}\sqrt{\mu _k}, \end{aligned}$$

respectively. The Laplacian energy of the graph G is defined as

$$\begin{aligned} \mathrm{LE}(G)=\sum ^n_{i=1}\Big |\mu _i-\frac{2m}{n}\Big |. \end{aligned}$$

In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.

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

Similar content being viewed by others

References

  1. Biler, P., Witkowski, A.: Problems in Mathematical Analysis. Marcel Dekker, Inc, New York (1990)

  2. Buckley, F., Harray, F.: Distance in Graphs. Addison-Wesley, Redwood (1990)

    Google Scholar 

  3. Das, K.C.: The Laplacian spectrum of a graph. Comput. Math. Appl. 48, 715–724 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Das, K.C.: A sharp upper bound for the number of spanning trees of a graph. Graphs Comb. 23, 625–632 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Das, K.C., Xu, K., Gutman, I.: Comparison between Kirchhoff index and Laplacian-energy-like invariant of graphs. Linear Algebra Appl. 436, 3661–3671 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Das, K.C., Gungar, A.D., Sinan Çevik, A.: On the Kirchhoff index and the resistance-distance energy of a graph. MATCH Commun. Math. Comput. Chem. 67(2), 541–556 (2012)

    MathSciNet  Google Scholar 

  7. Das, K.C.: On Kirchhoff Index of graphs. Zeitschrift für Naturforschung A 68(a), 531–538 (2013)

    Google Scholar 

  8. Das, K.C., Gutman, I., Çevik, A.S.: On the Laplacian-energy-like invariant. Linear Algebra Appl. 442, 58–68 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Das, K.C., Gutman, I., Çevik, A.S., Zhou, B.: On Laplacian energy. MATCH Commun. Math. Comput. Chem. 70(2), 689–696 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Das, K.C., Mojallal, S.A.: On Laplacian energy of graphs. Discrete Math. 325, 52–64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Estrada, E., Hatano, N.: Topological atomic displacements, Kirchhoff and Wiener indices of molecules. Chem. Phys. Lett. 486, 166–170 (2010)

    Article  Google Scholar 

  12. Gutman, I., Mohar, B.: The Quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982–985 (1996)

    Article  Google Scholar 

  13. Gutman, I., Zhou, B.: Laplacian energy of a graph. Linear Algebra Appl. 414, 29–37 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ilić, A., Krtinić, D., Ilić, M.: On Laplacian like energy of trees. MATCH Commun. Math. Comput. Chem. 64, 111–122 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Klein, D.J., Randić, M.: Resistance distance. Applied graph theory and discrete mathematics in chemistry (Saskatoon, SK, 1991). J. Math. Chem. 12(1–4), 81–95 (1993)

    Article  MathSciNet  Google Scholar 

  16. Li, J., Shiu, W.C., Chan, W.H.: The Laplacian spectral radius of some graphs. Linear Algebra Appl. 431, 99–103 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, B., Huang, Y., You, Z.: A survey on the Laplacian-energy-like invariant. MATCH Commun. Math. Comput. Chem. 66, 713–730 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Liu, J., Liu, B.: A Laplacian-energy-like invariant of a graph. MATCH Commun. Math. Comput. Chem. 59, 397–419 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Lu, M., Zhang, L.-Z., Tian, F.: Lower bounds of the Laplacian spectrum of graphs based on diameter. Linear Algebra Appl. 420, 400–406 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197, 198, 143–176 (1994)

  21. Pólya, G., Szegö, G.: Problems and Theorems in Analysis, Series, Integral Calculus, Theory of Functions, vol. I. Springer, Berlin (1972)

    MATH  Google Scholar 

  22. Song, Y.Z., Arbelaez, P., Hall, P., Li, C., Balikai, A.: Finding semantic structures in image hierarchies using Laplacian graph energy. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) Computer VisionCECV 2010, European Conference on Computer Vision, 2010, Part IV, pp. 694–707. Springer, Berlin (2010)

  23. Stevanović, D., Ilić, A., Onisor, C., Diudea, M.: LEL—a newly designed molecular descriptor. Acta Chim. Slov. 56, 410–417 (2009)

    Google Scholar 

  24. Xiao, W., Gutman, I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284–289 (2003)

    Article  Google Scholar 

  25. Xu, K., Das, K.C.: Extremal Laplacian-energy-like invariant of graphs with given matching number. Electron. J. Linear Algebra 26(1), 131–140 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, Y., Klein, D.J.: A recursion formula for resistance distances and its applications. Discrete Appl. Math. 161, 2702–2715 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, Y., Zhang, H., Klein, D.J.: New Nordhaus–Gaddum-type results for the Kirchhoff index. J. Math. Chem. 49, 1587–1598 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, B.X.: The Laplacian-energy like of graphs. Appl. Math. Lett. 24, 1604–1607 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhu, H.-Y., Klein, D.J., Lukovits, I.: Extensions of the Wiener number. J. Chem. Inf. Comput. Sci. 36, 420–428 (1996)

    Article  Google Scholar 

Download references

Acknowledgments

The authors are much grateful to two anonymous referees for their careful reading and valuable comments on our paper, which have considerably improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea

    Kinkar Ch. Das

  2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People’s Republic of China

    Kexiang Xu

Authors
  1. Kinkar Ch. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kexiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kinkar Ch. Das.

Additional information

Communicated by Xueliang Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, K.C., Xu, K. On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs. Bull. Malays. Math. Sci. Soc. 39 (Suppl 1), 59–75 (2016). https://doi.org/10.1007/s40840-015-0280-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-015-0280-4

Keywords

Mathematics Subject Classification

Search

Navigation