Skip to main content
SpringerLink
Log in

Inequalities on the generalized atom bond connectivity index

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

Abstract

The atom-bond connectivity and the generalized atom-bond connectivity indices have shown to be useful in the QSPR/QSAR researches. The aim of this paper is to obtain new inequalities for these indices and characterize graphs extremal with respect to them.

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. B. Bollobás, P. Erdős, Graphs of extremal weights. Ars Combin. 50, 225–233 (1998)

    Google Scholar 

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

    Google Scholar 

  3. X. Li, I. Gutman, Mathematical Aspects of Randić Type Molecular Structure Descriptors (Univ. Kragujevac, Kragujevac, 2006).

    Google Scholar 

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

    Google Scholar 

  5. M. Randić, On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975). https://doi.org/10.1021/ja00856a001

    Article  Google Scholar 

  6. X. Chen, K.C. Das, Solution to a conjecture on the maximum ABC index of graphs with given chromatic number. Discr. Appl. Math. 251, 126–134 (2018). https://doi.org/10.1016/j.dam.2018.05.063

    Article  Google Scholar 

  7. X. Chen, G. Hao, Extremal graphs with respect to generalized \(ABC\) index. Discr. Appl. Math. 243, 115–124 (2018). https://doi.org/10.1016/j.dam.2018.01.013

    Article  Google Scholar 

  8. KCh. Das, I. Gutman, B, , Furtula, On atom-bond connectivity index. Chem. Phys. Lett. 511, 452–454 (2011). https://doi.org/10.1016/j.cplett.2011.06.049

    Article  CAS  Google Scholar 

  9. B. Furtula, A. Graovac, D. Vukicević, Augmented Zagreb index. J. Math. Chem. 48, 370–380 (2010). https://doi.org/10.1007/s10910-010-9677-3

    Article  CAS  Google Scholar 

  10. I. Gutman, B. Furtula, M. Ivanović, Notes on trees with minimal atom bond connectivity index. MATCH Commun. Math. Comput. Chem. 67, 467–482 (2012)

    CAS  Google Scholar 

  11. H. Kober, On the arithmetic and geometric means and on Hölder’s inequality. Proc. Am. Math. Soc. 9, 452–459 (1958)

    Google Scholar 

  12. B. Zhou, I. Gutman, T. Aleksić, A note on Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 441–446 (2008)

    CAS  Google Scholar 

  13. R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees. MATCH Commun. Math. Comput. Chem. 64, 359–372 (2010)

    CAS  Google Scholar 

  14. I. Gutman, Multiplicative Zagreb indices of trees. Bull. Int. Math. Virt. Inst. 1, 13–19 (2011)

    Google Scholar 

  15. A. Iranmanesh, M.A. Hosseinzadeh, I. Gutman, On multiplicative Zagreb indices of graphs. Iran. J. Math. Chem. 3(2), 145–154 (2012). https://doi.org/10.22052/IJMC.2012.5234

    Article  Google Scholar 

  16. M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative versions of first Zagreb index. MATCH Commun. Math. Comput. Chem. 68, 217–230 (2012)

    Google Scholar 

  17. T. Popoviciu, Sur les équations algébriques ayant toutes leurs racines réelles. Mathematica (Cluj). 9, 129–145 (1935)

    Google Scholar 

  18. J. Szökefalvi Nagy, Über algebraische Gleichungen mit lauter reellen Wurzeln. Jahresber. der dtsch. mathematiker-Vereingung. 27, 37–43 (1918)

    Google Scholar 

  19. R. Sharma, M. Gupta, G. Kopor, Some better bounds on the variance with applications. J. Math. Ineq. 4(3), 355–367 (2010). https://doi.org/10.7153/jmi-04-32

    Article  Google Scholar 

  20. D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and new inequalities in analysis. Netherlands (1993). https://doi.org/10.1007/978-94-017-1043-5

    Article  Google Scholar 

  21. L.A. Székely, L.H. Clark, R.C. Entringer, An inequality for degree sequences. Discr. Math. 103, 293–300 (1992). https://doi.org/10.1016/0012-365X(92)90321-6

    Article  Google Scholar 

  22. J.M. Rodríguez, J.L. Sánchez, J.M. Sigarreta, On the first general Zagreb index. J. Math. Chem. 56, 1849–1864 (2018). https://doi.org/10.1007/s10910-017-0816-y

    Article  CAS  Google Scholar 

  23. B. Hollas, On the variance of topological indices that depend on the degree of a vertex. Match Commun. Math. Comput. Chem. 54, 341–350 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Chilpancingo, Mexico

    Ricardo Abreu-Blaya

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    Rosalio Reyes & José M. Rodríguez

  3. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650, Acalpulco Gro., Mexico

    José M. Sigarreta

Authors
  1. Ricardo Abreu-Blaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rosalio Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José M. Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José M. Sigarreta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rosalio Reyes.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Funding

This work was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abreu-Blaya, R., Reyes, R., Rodríguez, J.M. et al. Inequalities on the generalized atom bond connectivity index. J Math Chem 59, 775–791 (2021). https://doi.org/10.1007/s10910-021-01219-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-021-01219-5

Keywords

Mathematics Subject Classification

Search

Navigation