Skip to main content
SpringerLink
Log in

Further Inequalities Between Vertex-Degree-Based Topological Indices

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Topological indices play an important role in mathematical chemistry especially in the quantitative structure–property relationship and quantitative structure–activity relationship studies. Recent research indicates that the augmented Zagreb index (AZI) possess the best correlating ability among several topological indices to predict the certain physico-chemical properties of particular types of molecules. In the present work, several novel bounds (lower and upper) for the AZI in terms of first geometric–arithmetic index, Randić index, atom-bond connectivity index, sum-connectivity index, modified second Zagreb index and harmonic index are established. Moreover, the geometric–arithmetic index and modified second Zagreb index are also among the well known topological indices. Various relations between these two topological indices and some of the aforementioned indices are also derived.

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. Ali, A., Bhatti, A. A.: A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles. Kuwait J. Sci. 43(4) (2016)

  2. Ali, A., Raza, Z., Bhatti, A.A.: On the augmented Zagreb index. Kuwait J. Sci. 43(2), 48–63 (2016)

    MathSciNet  Google Scholar 

  3. Ali, A., Bhatti, A.A., Raza, Z.: The augmented Zagreb index, vertex connectivity and matching number of graphs. B. Iran. Math. Soc. 42(2), 417–425 (2016)

    MathSciNet  Google Scholar 

  4. Berrocal, L., Olivieri, A., Rada, J.: Extremal values of vertex-degree-based topological indices over hexagonal systems with fixed number of vertices. Appl. Math. Comput. 243, 176–183 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Das, K.C., Trinajstić, N.: Comparison between first geometric-arithmetic index and atom-bond connectivity index. Chem. Phys. Lett. 497, 149–151 (2010)

    Article  Google Scholar 

  6. Deng, H., Balachandran, S., Ayyaswamy, S.K., Venkatakrishnan, Y.B.: On the harmonic index and the chromatic number of a graph. Discrete Appl. Math. 161, 2740–2744 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  8. Estrada, E., Torres, L., Rodríguez, L., Gutman, I.: An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. A 37, 849–855 (1998)

    Google Scholar 

  9. Fajtlowicz, S.: On conjectures of Graffiti-II. Congr. Numer. 60, 187–197 (1987)

    MathSciNet  MATH  Google Scholar 

  10. Furtula, B., Graovac, A., Vukičević, D.: Augmented Zagreb index. J. Math. Chem. 48, 370–380 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Furtula, B., Gutman, I., Dehmer, M.: On structure-sensitivity of degree-based topological indices. Appl. Math. Comput. 219, 8973–8978 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Ghorbani, M., Songhori, M.: Computing multiplicative zagreb indices with respect to chromatic and clique numbers. Iranian J. Math. Chem. 5, 11–18 (2014)

    Google Scholar 

  13. Goubko, M., Gutman, I.: Degree-based topological indices: optimal trees with given number of pendents. Appl. Math. Comput. 240, 387–398 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Gutman, I.: Degree-based topological indices. Croat. Chem. Acta 86(4), 351–361 (2013)

    Article  Google Scholar 

  15. Gutman, I., Furtula, B. (eds.): Novel Molecular Structure Descriptors Theory and Applications. University of Kragujevac, Kragujevac (2010)

    Google Scholar 

  16. Gutman, I., Furtula, B.: Vertex-degree-based molecular structure descriptors of benzenoid systems and phenylenes. J. Serb. Chem. Soc. 77, 1031–1036 (2012)

    Article  Google Scholar 

  17. Gutman, I., Tošović, J.: Testing the quality of molecular structure descriptors: vertex-degree-based topological indices. J. Serb. Chem. Soc. 78, 805–810 (2013)

    Article  Google Scholar 

  18. Gutman, I., Zhong, L., Xu, K.: Relating the \(ABC\) and harmonic indices. J. Serb. Chem. Soc. 79(5), 557–563 (2014)

    Article  Google Scholar 

  19. Gutman, I., Furtula, B., Elphick, C.: Three new/old vertex-degree-based topological indices. MATCH Commun. Math. Comput. Chem. 72, 617–632 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Harary, F.: Graph Theory. Addison-Wesley, Reading, MA (1969)

    MATH  Google Scholar 

  21. Hollas, B.: The covariance of topological indices that depend on the degree of a vertex. MATCH Commun. Math. Comput. Chem. 54(1), 177–187 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Huang, Y., Liu, B., Gan, L.: Augmented Zagreb index of connected graphs. MATCH Commun. Math. Comput. Chem. 67, 483–494 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Karcher, W., Devillers, J.: Practical Applications of Quantitative Structure–Activity Relationships (QSAR) in Environmental Chemistry and Toxicology. Springer, Berlin (1990)

    Google Scholar 

  24. Miličević, A., Nikolić, S.: On variable Zagreb indices. Croat. Chem. Acta 77, 97–101 (2004)

    Google Scholar 

  25. Nikolić, S., Kovačević, G., Miličević, A., Trinajstić, N.: The zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)

    Google Scholar 

  26. Palacios, J.L.: A resistive upper bound for the ABC index. MATCH Commun. Math. Comput. Chem. 72, 709–713 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Rada, J., Cruz, R., Gutman, I.: Benzenoid systems with extremal vertex-degree-based topological indices. MATCH Commun. Math. Comput. Chem. 72, 125–136 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Randić, M.: On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)

    Article  Google Scholar 

  29. Raza, Z., Bhatti, A.A., Ali, A.: More on comparison between first geometric-arithmetic index and atom-bond connectivity index. Miskolc Math. Notes 17(1), 561–570 (2016)

  30. Vukičević, D., Furtula, B.: Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46, 1369–1376 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  32. Vukičević, D., Durdević, J.: Bond additive modeling 10. Upper and lower bounds of bond incident degree indices of catacondensed fluoranthenes. Chem. Phys. Lett. 515, 186–189 (2011)

    Article  Google Scholar 

  33. Vukičević, Ẑ.K., Popivoda, G.: Chemical trees with extreme values of Zagreb indices and coindices. Iranian J. Math. Chem. 5, 19–29 (2014)

    Google Scholar 

  34. Wang, S., Zhou, B., Trinajstić, N.: On the sum-connectivity index. Filomat 25(3), 29–42 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, D., Huang, Y., Liu, B.: Bounds on augmented Zagreb index. MATCH Commun. Math. Comput. Chem. 68, 209–216 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Xu, X.: Relationships between harmonic index and other topological indices. Appl. Math. Sci. 6, 2013–2018 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Zhong, L., Xu, K.: Inequalities between vertex-degree-based topological Indices. MATCH Commun. Math. Comput. Chem. 71, 627–642 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Zhou, B., Trinajstić, N.: On a novel connectivity index. J. Math. Chem. 46, 1252–1270 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhou, B., Trinajstić, N.: Relations between the product and sum-connectivity indices. Croat. Chem. Acta 85, 363–365 (2012)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referees for reviewing the manuscript twice and for their insightful comments and valuable suggestions, which led to a number of improvements in the earlier versions of the manuscript.

Author information

Authors and Affiliations

  1. Department of Sciences and Humanities, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore, Pakistan

    Akbar Ali, Akhlaq Ahmad Bhatti & Zahid Raza

  2. Department of Mathematics, College of Science, University of Sharjah, Sharjah, UAE

    Zahid Raza

Authors
  1. Akbar Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akhlaq Ahmad Bhatti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zahid Raza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akbar Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, A., Bhatti, A.A. & Raza, Z. Further Inequalities Between Vertex-Degree-Based Topological Indices. Int. J. Appl. Comput. Math 3, 1921–1930 (2017). https://doi.org/10.1007/s40819-016-0213-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0213-4

Keywords

Mathematics Subject Classification

Search

Navigation