On resistance-distance and Kirchhoff index

Original Paper

Abstract

We provide some properties of the resistance-distance and the Kirchhoff index of a connected (molecular) graph, especially those related to its normalized Laplacian eigenvalues.

Keywords

Kirchhoff index Laplacian eigenvalues Laplacian matrix Resistance-distance 

References

  1. 1.
    Klein D.J., Randić M.: Resistance distance. J. Math. Chem. 12, 81 (1993)CrossRefGoogle Scholar
  2. 2.
    Kirchhoff G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, 497 (1847)CrossRefGoogle Scholar
  3. 3.
    Edmister J.A.: Electric Circuits. McGraw-Hill, New York (1965)Google Scholar
  4. 4.
    Johnson D.E., Johnson J.R.: Graph Theory with Engineering Applications. Ronald Press Co., New York (1972)Google Scholar
  5. 5.
    Seshu S., Reed M.B.: Linear Graphs and Electrical Networks. Addison-Wesley, Reading (1961)Google Scholar
  6. 6.
    R.M. Foster, The average impedance of an electrical network. in Contributions to Applied Mechanics (Reissner Aniversary Volume, Edwards Brothers, Ann Arbor, 1949), pp. 333–340Google Scholar
  7. 7.
    Palacios J.L.: Forster’s formulas via probability and the Kirchhoff index. Methodol. Comput. Appl. Probab. 6, 381 (2004)CrossRefGoogle Scholar
  8. 8.
    Bonchev D., Balaban A.T., Liu X., Klein D.J.: Molecular cyclicityand centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances. Int. J. Quantum Chem. 50, 1 (1994)CrossRefGoogle Scholar
  9. 9.
    L. Lovász, Random walks on graphs: a survey. in Combinatorics, Paul Erdös is eighty, vol. 2 (János Bolyai Mathematical Society, Budapest, 1996), pp. 353–397Google Scholar
  10. 10.
    Zhu H.Y., Klein D.J., Lukovits I.: Extensions of the Wiener number. J. Chem. Inf. Comput. Sci. 36, 420 (1996)Google Scholar
  11. 11.
    Gutman I., Mohar B.: The quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982 (1996)Google Scholar
  12. 12.
    Lukovits L., Nikolić S., Trinajstić N.: Resistance distance in regular graphs. Int. J. Quantum Chem. 71, 217 (1999)CrossRefGoogle Scholar
  13. 13.
    Babić D., Klein D.J., Lukovits I., Nikolić S., Trinajstić N.: Resistance-distance matrix. A computational algorithm and its applications. Int. J. Quantum Chem. 90, 166 (2002)CrossRefGoogle Scholar
  14. 14.
    Xiao W., Gutman I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284 (2003)Google Scholar
  15. 15.
    Klein D.J., Palacios J.L., Randić M., Trinajstić N.: Random walks and chemical graph theory. J. Chem. Inf. Comput. Sci. 44, 1521 (2004)Google Scholar
  16. 16.
    Chen H., Zhang F.: Resistance distance and the normalized Laplacian spectrum. Discret. Appl. Math. 155, 654 (2007)CrossRefGoogle Scholar
  17. 17.
    Zhou B., Trinajstić N.: A note on Kirchhoff index. Chem. Phys. Lett. 445, 120 (2008)CrossRefGoogle Scholar
  18. 18.
    D. Janežič, A. Miličević, S. Nikolić, N. Trinajstić, Graph Theoretical Matrices in Chemistry (Mathematical Chemistry Monographs No. 3, University of Kragujevac, Kragujevac, 2007)Google Scholar
  19. 19.
    Merris R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197(198), 143 (1994)CrossRefGoogle Scholar
  20. 20.
    Trinajstić N., Babić D., Nikolić S., Plavšić D., Amić D., Mihalić Z.: Laplacian matrix in chemistry. J. Chem. Inf. Comput. Sci. 34, 368 (1994)Google Scholar
  21. 21.
    Chung F.R.K.: Spectral Graph Theory. American MathematicalSociety, Providence (1997)Google Scholar
  22. 22.
    Mohar B., Babić D., Trinajstić N.: A novel definition of the Wiener index for trees. J. Chem. Inf. Comput. Sci. 33, 153 (1993)Google Scholar
  23. 23.
    Horn R.A., Johnson C.R.: Matrix Analysis, pp. 176–178. Cambridge University Press, Cambridge (1985)Google Scholar
  24. 24.
    Cvetković D.M., Doob M., Sachs H.: Spectra of Graphs —Theory and Application, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsSouth China Normal UniversityGuangzhouChina
  2. 2.The Rugjer Bošković InstituteZagrebCroatia

Personalised recommendations