Skip to main content
SpringerLink
Log in

Mostar index

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic graphs. Then we show how it can be efficiently computed for large classes of chemically interesting graphs using a variant of the cut method introduced by Klavžar, Gutman and Mohar. Explicit formulas are presented for several classes of benzenoid graphs and Cartesian products. At the end we state several conjectures and list some open problems.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. H. Abdo, S. Brandt, D. Dimitrov, The total irregularity of a graph. Discrete Math. Theor. Comput. Sci. 16, 201–206 (2014)

    Google Scholar 

  2. M.O. Albertson, The irregularity of a graph. Ars Comb. 46, 219–225 (1997)

    Google Scholar 

  3. K. Balakrishnan, B. Brešar, M. Chagat, S. Klavžar, A. Vesel, P. Žigert Pleteršek, Equal opportunity networks, distance-balanced graphs, and Wiener game. Discrete Optim. 12, 150–154 (2014)

    Article  Google Scholar 

  4. T. Došlić, B. Furtula, A. Graovac, I. Gutman, S. Moradi, Z. Yarahmadi, On vertex-degree-based molecular structure descriptors. MATCH Commun. Math. Comput. Chem. 66, 613–626 (2011)

    Google Scholar 

  5. T. Došlić, T. Reti, D. Vukičević, On vertex degree indices of connected graphs. Chem. Phys. Lett. 512, 283–286 (2011)

    Article  CAS  Google Scholar 

  6. T. Došlić, Vertex-weighted Wiener polynomials for composite graphs. Ars Math. Contemp. 1, 66–80 (2008)

    Article  Google Scholar 

  7. D. Djoković, Distance preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14, 263–267 (1973)

    Article  Google Scholar 

  8. I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes New York XXVI I, 9–15 (1994)

    Google Scholar 

  9. I. Gutman, A. Dobrynin, The Szeged index: a success story. Graph Theory Notes New York 34(37–44), 37–44 (1998)

    Google Scholar 

  10. I. Gutman, Degree-based topological indices. Croat. Chem. Acta 86, 351–361 (2013)

    Article  CAS  Google Scholar 

  11. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)

    Article  CAS  Google Scholar 

  12. F. Harary, Graph Theory (Addison-Wesley, Reading, 1969)

    Book  Google Scholar 

  13. A. Ilić, S. Klavžar, M. Milanović, On distance-balanced graphs. Eur. J. Comb. 31, 733–737 (2010)

    Article  Google Scholar 

  14. J. Jerebic, S. Klavžar, D.F. Rall, Distance-balanced graphs. Ann. Comb. 12, 71–79 (2008)

    Article  Google Scholar 

  15. S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relations. J. Chem. Inf. Comput. Sci. 35, 590–593 (1995)

    Article  Google Scholar 

  16. S. Klavžar, A bird’s eye view of the cut method and a survey of its applications in chemical graph theory. MATCH Commun. Math. Comput. Chem. 60, 255–274 (2008)

    Google Scholar 

  17. K. Kutnar, A. Malnič, D. Marušič, Š. Miklavič, Distance-balanced graphs: symmetry conditions. Discrete Math. 306, 1881–1894 (2006)

    Article  Google Scholar 

  18. V. Latora, M. Marchiori, Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001)

    Article  CAS  PubMed  Google Scholar 

  19. Š. Miklavič, P. Šparl, \(\ell \)-distance-balanced graphs. Discrete Appl. Math. 244, 143–154 (2018)

    Article  Google Scholar 

  20. M. Randić, Molecular bonding profiles. J. Math. Chem. 19, 375–392 (1996)

    Article  Google Scholar 

  21. R. Sharafdini, T. Réti, On the transmission-based graph topological indices. arXiv:1710.08176v1

  22. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, 2nd edn. (Wiley-VCH, Weinheim, 2009)

    Google Scholar 

  23. D. Vukičević, M. Gašperov, Bond additive modelling 1. Adriatic indices. Croat. Chem. Acta 83, 243–260 (2010)

    Google Scholar 

  24. D. Vukičević, Bond Additive Modelling 2. Mathematical properties of Max-min rodeg index. Croat. Chem. Acta 83, 261–273 (2010)

    Google Scholar 

  25. D. Vukičević, Bond additive modelling 4. QSPR and QSAR studies of the variable Adriatic indices. Croat. Chem. Acta 84, 87–91 (2011)

    Article  CAS  Google Scholar 

  26. D.B. West, Introduction to Graph Theory (Prentice-Hall, Upper Saddle River, 1996)

    Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  28. P. Winkler, Isometric embeddings in products of complete graphs. Discrete Appl. Math. 7, 221–225 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

This work has been supported in part by Croatian Science Foundation under the Project LightMol (Grant No. IP-2016-06-1142). Also, the authors gratefully acknowledge partial support from Croatian-Slovenian bilateral project Modeling adsorption on nanostructures: Graph-theoretic approach. The research was initiated at the workshop Applications of Graph Theory in Automatization, Robotics, and Computing held at the Faculty of Mechanical Engineering and Computing in Mostar that brought together all authors.

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia

    Tomislav Došlić

  2. Faculty of Sciences, University of Zagreb, Zagreb, Croatia

    Ivica Martinjak

  3. FMF, University of Ljubljana, Ljubljana, Slovenia

    Riste Škrekovski

  4. Faculty of Information Studies, Novo Mesto, Slovenia

    Riste Škrekovski

  5. FAMNIT, University of Primorska, Koper, Slovenia

    Riste Škrekovski

  6. Faculty of Sciences and Education, University of Mostar, Mostar, Bosnia and Herzegovina

    Sanja Tipurić Spužević

  7. Faculty of Mechanical Engineering and Computing, University of Mostar, Mostar, Bosnia and Herzegovina

    Ivana Zubac

Authors
  1. Tomislav Došlić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivica Martinjak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Riste Škrekovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sanja Tipurić Spužević
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ivana Zubac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomislav Došlić.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Došlić, T., Martinjak, I., Škrekovski, R. et al. Mostar index. J Math Chem 56, 2995–3013 (2018). https://doi.org/10.1007/s10910-018-0928-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-018-0928-z

Keywords

Search

Navigation