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Journal of Mathematical Chemistry

, Volume 46, Issue 4, pp 1377–1393 | Cite as

Bounds on Harary index

  • Kinkar Ch. DasEmail author
  • Bo Zhou
  • N. Trinajstić
Original Paper

Abstract

In this paper, we obtain the lower and upper bounds on the Harary index of a connected graph (molecular graph), and, in particular, of a triangle- and quadrangle-free graphs in terms of the number of vertices, the number of edges and the diameter. We give the Nordhaus–Gaddum-type result for Harary index using the diameters of the graph and its complement. Moreover, we compare Harary index and reciprocal complementary Wiener number for graphs.

Keywords

Harary index Triangle-free graphs Quadrangle-free graphs Diameter Lower bound Upper bound 

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References

  1. 1.
    Plavšić D., Nikolić S., Trinajstić N., Mihalić Z.: On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12, 235–250 (1993)CrossRefGoogle Scholar
  2. 2.
    Ivanciuc O., Balaban T.S., Balaban A.T.: Reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem. 12, 309–318 (1993)CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Hosoya H.: Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332–2339 (1971)Google Scholar
  5. 5.
    Wiener H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  6. 6.
    Ivanciuc O., Ivanciuc T., Balaban A.T.: Design of topological indices. Part 10. Parameters based on electronegativity and vovalent radius for the computation of molecular graph descriptors for heteroatom-containing molecules. J. Chem. Inf. Comput. Sci. 38, 395–495 (1998)Google Scholar
  7. 7.
    Diudea M.V.: Indices of reciprocal properties or Harary indices. J. Chem. Inf. Comput. Sci. 37, 292–299 (1997)Google Scholar
  8. 8.
    Lučić B., Miličević A., Nikolić S., Trinajstić N.: Harary index-twelve years later. Croat. Chem. Acta 75, 847–868 (2002)Google Scholar
  9. 9.
    Devillers, J., Balaban A.T. (eds). Topological Indices and Related Descriptors in QSAR and QSPR. Gordon & Breach, Amsterdam (1999)Google Scholar
  10. 10.
    Todeschini R., Consonni V.: Handbook of Molecular Descriptors. Weinheim, Wiley-VCH (2000)CrossRefGoogle Scholar
  11. 11.
    Mihalić Z., Trinajstić N.: A graph-theoretical approach to structure-property relationships. J. Chem. Educ. 69, 701–712 (1999)CrossRefGoogle Scholar
  12. 12.
    Ivanciuc O.: QSAR comparative study of Wiener descriptors for weighted molecular graphs. J. Chem. Inf. Comput. Sci. 40, 1412–1422 (2000)Google Scholar
  13. 13.
    Trinajstić N., Nikolić S., Basak S.C., Lukovits I.: Distance indices and their hyper-counterparts: intercorrelation and use in the structure-property modeling. SAR QSAR Environ. Res. 12, 31–54 (2001)CrossRefGoogle Scholar
  14. 14.
    Zhou B., Cai X., Trinajstić N.: On Harary index. J. Math. Chem. 44, 611–618 (2008)CrossRefGoogle Scholar
  15. 15.
    B. Zhou, X. Cai, N. Trinajstić, On reciprocal complementary Wiener number. Discrete Appl. Math. (in press). doi: 10.1016/j.dam.2008.09.010
  16. 16.
    Trinajstić N.: Chemical Graph Theory, 2nd revised edn. CRC Press, Boca Raton (1992)Google Scholar
  17. 17.
    Gutman I., Trinajstić N.: Graph theory and molecular orbitals. III. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)Google Scholar
  18. 18.
    Gutman I., Ruščić B., Trinajstić N., Wilcox C.F. Jr.: Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys. 62, 3399–3405 (1975)CrossRefGoogle Scholar
  19. 19.
    Nikolić S., Kovačević G., Mihalić A., Trinajstić N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)Google Scholar
  20. 20.
    Gutman I., Das K.C.: first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem. 50, 83–92 (2004)Google Scholar
  21. 21.
    Zhou B., Stevanović D.: A note on Zagreb indices. MATCH Commun. Math. Comput. Chem. 56, 571–578 (2006)Google Scholar
  22. 22.
    K.C. Das, I. Gutman, B. Zhou, New upper bounds on Zagreb indices. J. Math. Chem. doi: 10.1007/s10910-008-9475-3
  23. 23.
    Cvetković D.M., Doob M., Sachs H.: Spectra of Graphs-Theory and Application. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonRepublic of Korea
  2. 2.Department of MathematicsSouth China Normal UniversityGuangzhouPeople’s Republic of China
  3. 3.The Rugjer Bošković InstituteZagrebCroatia

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