Skip to main content
SpringerLink
Log in

Wiener index of certain families of hexagonal chains

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

The Wiener index is a topological index of a molecule, defined as the sum of distances between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. Hexagonal chains consist of hexagonal rings connected with each other by edges. This class of graphs contains molecular graphs of unbranched catacondensed benzenoid hydrocarbons. A segment of a chain is its maximal subchain with linear connected hexagons. Chains with segments of equal lengths can be coded by binary words. Formulas for the sums of Wiener indices of hexagonal chains of some families are derived and computational examples are presented.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Balaban, A.T.: Chemical graphs. L. Symmetry and enumeration of fibonacenes (unbranched catacondensed benzenoids isoarithmic with helicenes and zigzag catafusenes. MATCH Commun. Math. Comput. Chem. 24, 9–38 (1989)

    MATH  Google Scholar 

  2. Balaban, A.T., Motoc, I., Bonchev, D., Mekenyan, O.: Topological indices for structure–activity correlations. Top. Curr. Chem. 114, 21–55 (1983)

    Article  Google Scholar 

  3. Canfield, E.R., Robinson, R.W., Rouvray, D.H.: Determination of the Wiener molecular branching index for the general tree. J. Comput. Chem. 6, 598–609 (1985)

    Article  MathSciNet  Google Scholar 

  4. Dehmer, M., Emmert-Streib, F. (eds.): Quantitative Graph Theory: Mathematical Foundations and Applications, Discrete Mathematics and Its Applications. Chapman and Hall/CRC, London (2014)

    Google Scholar 

  5. Devillers, J., Balaban, A.T. (eds.): Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Reading (1999)

    Google Scholar 

  6. Diudea, M.V. (ed.): QSPR/QSAR Studies by Molecular Descriptors. Nova, Huntington (2001)

    Google Scholar 

  7. Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index for trees: theory and applications. Acta Appl. Math. 66(3), 211–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dobrynin, A.A., Gutman, I., Klavžar, S., Žigert, P.: Wiener index of hexagonal systems. Acta Appl. Math. 72(3), 247–294 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dobrynin, A.A.: On the Wiener index of fibonacenes. MATCH Commun. Math. Comput. Chem. 64(3), 707–726 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Dobrynin, A.A.: On the Wiener index of certain families of fibonacenes. MATCH Commun. Math. Comput. Chem. 70(2), 565–574 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Dobrynin, A.A.: Wiener index of hexagonal chains with segments of equal length. In: Dehmer, M., Emmert-Streib, F. (eds.) Quantitative Graph Theory: Mathematical Foundations and Applications, Discrete Mathematics and Its Applications, pp. 81–109. Chapman and Hall/CRC, London (2014)

    Chapter  Google Scholar 

  12. Entringer, R.C.: Distance in graphs: trees. J. Combin. Math. Combin. Comput. 24, 65–84 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Gutman, I.: Topological properties of benzenoid systems. Topics Curr. Chem. 162, 21–28 (1992)

    Google Scholar 

  14. Gutman, I., Cyvin, S.J.: Introduction to the Theory of Benzenoid Hydrocarbons. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  15. Gutman, I., Klavžar, S.: Chemical graph theory of fibonacenes. MATCH Commun. Math. Comput. Chem. 55, 39–54 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  17. Gutman, I., Yeh, Y.N., Lee, S.L., Luo, Y.L.: Some recent results in the theory of the Wiener number. Indian J. Chem. 32A, 651–661 (1993)

    Google Scholar 

  18. Gutman, I., Furtula, B., (eds.): Distance in Molecular Graphs—Theory. Mathematical Chemistry Monographs, 12. University of Kragujevac and Faculty of Science Kragujevac, Kragujevac (2012)

  19. Gutman, I., B. Furtula, B., (eds.): Distance in Molecular Graphs—Applications, Mathematical Chemistry Monographs, 13. University of Kragujevac and Faculty of Science Kragujevac, Kragujevac (2012)

  20. Knor, M., Škrekovski, R., Tepeh, A.: Mathematical aspects of Wiener index. Ars Math. Contemp. 11(2), 327–352 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nikolić, S., Trinajstić, N., Mihalić, Z.: The Wiener index: developments and applications. Croat. Chem. Acta 68, 105–129 (1995)

    Google Scholar 

  22. Polansky, O.E., Bonchev, D.: The Wiener number of graphs. I. General theory and changes due to some graph operations. MATCH Commun. Math. Comput. Chem. 21, 133–186 (1986)

    MathSciNet  MATH  Google Scholar 

  23. Rouvray, D.H.: Should we have designs on topological indices? In: King, R.B. (ed.) Chemical Application of Topology and Graph Theory, pp. 159–177. Elsevier, Amsterdam (1983)

    Google Scholar 

  24. Rouvray, D.H.: The modeling of chemical phenomena using topological indices. J. Comput. Chem. 8, 470–480 (1987)

    Article  MathSciNet  Google Scholar 

  25. Todeschini, R., Consonni, V.: Handbook of Molecular Descriptors. Wiley-VCH, Weinheim (2000)

    Book  Google Scholar 

  26. Trinajstić, N.: Chemical Graph Theory. CRC Press, Boca Raton (1983, 1992)

  27. Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Andrey A. Dobrynin

  2. Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

    Ehsan Estaji

Authors
  1. Andrey A. Dobrynin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehsan Estaji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey A. Dobrynin.

Additional information

This work was supported by the Russian Foundation for Basic Research (Project Nos. 16-01-00499, 17-51-560008), Iranian National Science Foundation (Project No. 96004167) and the Program of fundamental scientific researches of the SB RAS I.5.1 (Project No. 0314-2016-0016).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dobrynin, A.A., Estaji, E. Wiener index of certain families of hexagonal chains. J. Appl. Math. Comput. 59, 245–256 (2019). https://doi.org/10.1007/s12190-018-1177-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-018-1177-9

Keywords

Mathematics Subject Classification

Search

Navigation