Skip to main content
SpringerLink
Log in

Applications of the inverse degree index to molecular structures

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

Abstract

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs which are extremal with respect to them. In particular, we obtain several inequalities relating the inverse degree index with the first and second Zagreb indices, the general first and second Zagreb indices, the Forgotten index, the general sum-connectivity index, the Sombor index and the misbalance indeg index, and several parameters of the molecular graph as the number of vertices, the number of edges, the minimum degree and the maximum degree. Also, we compute the inverse degree index for some classes of chemical graphs. Furthermore, some applications are given to the study of the physicochemical properties of three classes of compounds: polyaromatic hydrocarbons, polychlorobiphenyls, and octane isomers.

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

Similar content being viewed by others

Availability of data and materials

Some data in Table 1 were extracted from [18, 19, 31]. The remaining data in Table 1 were obtained by calculation. The data on physicochemical properties of the studied compounds were obtained from [13, 41, 42].

References

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

    Google Scholar 

  2. V. Andova, M. Petrusevski, Variable Zagreb Indices and Karamata’s Inequality. MATCH Commun. Math. Comput. Chem. 65, 685–690 (2011)

    Google Scholar 

  3. B. Borovicanin, B. Furtula, On extremal Zagreb indices of trees with given domination number. Appl. Math. Comput. 279, 208–218 (2016)

    Google Scholar 

  4. G. Britto Antony Xavier, E. Suresh, I. Gutman, Counting relations for general Zagreb indices. Kragujevac J. Math. 38, 95–103 (2014)

    Google Scholar 

  5. P. Dankelmann, A. Hellwig, L. Volkmann, Inverse degree and edge-connectivity. Discret. Math. 309, 2943–2947 (2008)

    Google Scholar 

  6. K.C. Das, On comparing Zagreb indices of graphs. MATCH Commun. Math. Comput. Chem. 63, 433–440 (2010)

    Google Scholar 

  7. K.C. Das, K. Xu, J. Wang, On inverse degree and topological indices of graphs. Filomat 30(8), 2111–2120 (2016)

    Google Scholar 

  8. Entringer, R.: Bounds for the average distance-inverse degree product in trees. In: Combinatorics, Graph Theory, and Algorithms, vol. I, II, pp. 335–352. Kalamazoo, MI (1996)

  9. P. Erdös, J. Pach, J. Spencer, On the mean distance between points of a graph. Congr. Numer. 64, 121–124 (1988)

    Google Scholar 

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

    Google Scholar 

  11. B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices. Discret. Appl. Math. 178, 83–88 (2014)

    Google Scholar 

  12. B. Furtula, I. Gutman, A forgotten topological index. J. Math. Chem. 53(4), 1184–1190 (2015)

    CAS  Google Scholar 

  13. P. Gramatica, N. Navas, R. Todeschini, 3D-modelling and prediction by WHIM descriptors. Part 9. Chromatographic relative retention time and physico-chemical properties of polychlorinated biphenyls (PCBs). Chemom. Intell. Lab. Syst. 40, 53–63 (1998)

    CAS  Google Scholar 

  14. I. Gutman, Geometric approach to degree-based topological indices: Sombor indices. MATCH Commun. Math. Comput. Chem. 86(1), 11–16 (2021)

    Google Scholar 

  15. I. Gutman, B. Furtula (eds.), Recent Results in the Theory of Randić Index (University of Kragujevac, Kragujevac, 2008)

    Google Scholar 

  16. I. Gutman, J. Tosovic, Testing the quality of molecular structure descriptors. Vertex–degree–based topological indices. J. Serb. Chem. Soc. 78(6), 805–810 (2013)

    CAS  Google Scholar 

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

    CAS  Google Scholar 

  18. V.R. Kulli, Revan indices of oxide and honeycomb networks. Int. J. Math. Appl. 5(4–E), 663–667 (2017)

    Google Scholar 

  19. V.R. Kulli, F-Revan index and F-Revan polynomial of some families of Bnzenoid systems. J. Glob. Res. Math. Arch. 5(1), 1–6 (2018)

    Google Scholar 

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

    Google Scholar 

  21. X. Li, Y. Shi, A survey on the Randić index. MATCH Commun. Math. Comput. Chem. 59, 127–156 (2008)

    CAS  Google Scholar 

  22. X. Li, H. Zhao, Trees with the first smallest and largest generalized topological indices. MATCH Commun. Math. Comput. Chem. 50, 57–62 (2004)

    Google Scholar 

  23. X. Li, J. Zheng, A unified approach to the extremal trees for different indices. MATCH Commun. Math. Comput. Chem. 54, 195–208 (2005)

    CAS  Google Scholar 

  24. M. Liu, A simple approach to order the first Zagreb indices of connected graphs. MATCH Commun. Math. Comput. Chem. 63, 425–432 (2010)

    Google Scholar 

  25. M. Liu, B. Liu, Some properties of the first general Zagreb index. Australas. J. Comb. 47, 285–294 (2010)

    Google Scholar 

  26. V. Lokesha, K. Zeba Yasmeen, B. Chaluvaraju, T. Deepika, Computation of misbalance type degree indices of certain classes of derived-regular graph. Int. J. Math. Comb. 3, 61–69 (2019)

    Google Scholar 

  27. B. Lučić, N. Trinajstić, B. Zhou, Comparison between the sum-connectivity index and product-connectivity index for benzenoid hydrocarbons. Chem. Phys. Lett. 475, 146–148 (2009)

    Google Scholar 

  28. B. Lučić, S. Nikolić, N. Trinajstić, B. Zhou, S. Ivaniš Turk, Sum-connectivity index, in Novel Molecular Structure Descriptors-Theory and Applications I. ed. by I. Gutman, B. Furtula (University of Kragujevac, Kragujevac, 2010), pp.101–136

    Google Scholar 

  29. A. Martínez-Pérez, J.M. Rodríguez, J.M. Sigarreta, A new approximation to the geometric-arithmetic index. J. Math. Chem. 56(7), 1865–1883 (2018)

    Google Scholar 

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

    Google Scholar 

  31. Nandargi, M.R., Kulli, V.R.: The \((a,b)\)–Nirmala index. IJESRT 11(2) (2022)

  32. S. Nikolić, A. Miličević, N. Trinajstić, A. Jurić, On use of the variable Zagreb \(^\nu M_2\) index in QSPR: boiling points of benzenoid hydrocarbons. Molecules 9, 1208–1221 (2004)

    PubMed  PubMed Central  Google Scholar 

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

    Google Scholar 

  34. M. Randić, Novel graph theoretical approach to heteroatoms in QSAR. Chemom. Intell. Lab. Syst. 10, 213–227 (1991)

    Google Scholar 

  35. M. Randić, On computation of optimal parameters for multivariate analysis of structure-property relationship. J. Chem. Inf. Comput. Sci. 31, 970–980 (1991)

    Google Scholar 

  36. M. Randić, D. Plavšić, N. Lerš, Variable connectivity index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 41, 657–662 (2001)

    PubMed  Google Scholar 

  37. I. Redžepović, Chemical applicability of Sombor indices. J. Serb. Chem. Soc. 86(5), 445–457 (2021)

    Google Scholar 

  38. J.A. Rodríguez-Velázquez, J.M. Sigarreta, On the Randić index and conditional parameters of a graph. MATCH Commun. Math. Comput. Chem. 54, 403–416 (2005)

    Google Scholar 

  39. J.A. Rodríguez-Velázquez, J. Tomás-Andreu, On the Randić index of polymeric networks modelled by generalized Sierpinski graphs. MATCH Commun. Math. Comput. Chem. 74, 145–160 (2015)

    Google Scholar 

  40. M. Singh, KCh. Das, S. Gupta, A.K. Madan, Refined variable Zagreb indices: highly discriminating topological descriptors for QSAR/QSPR. Int. J. Chem. Model. 6(2–3), 403–428 (2014)

    Google Scholar 

  41. R. Todeschini, P. Gramatica, E. Marengo, R. Provenzani, Weighted holistic invariant molecular descriptors. Part 2. Theory development and applications on modeling physico-chemical properties of polyaromatic hydrocarbons (PAH). Chemom. Intell. Lab. Syst. 27, 221–229 (1995)

    CAS  Google Scholar 

  42. K. Tuppurainen, J. Ruuskanen, Electronic eigenvalue (EEVA): a new QSAR/QSPR descriptor for electronic substituent effects based on molecular orbital energies. A QSAR approach to the Ah receptor binding affinity of polychlorinated biphenyls (PCBs), dibenzo-pdioxins (PCDDs) and dibenzofurans (PCDFs). Chemosphere 41, 843–4848 (2000)

    CAS  PubMed  Google Scholar 

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

    Google Scholar 

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

    CAS  PubMed  Google Scholar 

  45. S. Zhang, W. Wang, T.C.E. Cheng, Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun. Math. Comput. Chem. 55, 579–592 (2006)

    Google Scholar 

  46. H. Zhang, S. Zhang, Unicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun. Math. Comput. Chem. 55, 427–438 (2006)

    CAS  Google Scholar 

  47. Z. Zhang, J. Zhang, X. Lu, The relation of matching with inverse degree of a graph. Discret. Math. 301, 243–246 (2005)

    Google Scholar 

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

    CAS  Google Scholar 

  49. B. Zhou, N. Trinajstić, On general sum-connectivity index. J. Math. Chem. 47, 210–218 (2010)

    CAS  Google Scholar 

Download references

Acknowledgements

This publication is part of the grant PID2019-106433GB-I00 funded by MCIN/AEI/ 10.13039/501100011033. J.M.R. is also supported by a grant from the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

We would like to thank the referees for their careful reading of the manuscript.

Funding

Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.

Author information

Author notes

  1. Edil D. Molina, José M. Rodríguez, José L. Sánchez, José M. Sigarreta contributed equally to this work.

Authors and Affiliations

  1. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame no. 54, 39650, Acapulco, Guerrero, Mexico

    Edil D. Molina, José L. Sánchez & José M. Sigarreta

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

    José M. Rodríguez

Authors
  1. Edil D. Molina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José M. Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José L. Sánchez
    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

Contributions

The authors contributed equally to this work.

Corresponding author

Correspondence to José M. Rodríguez.

Ethics declarations

Competing interest

The authors confirm that the content of this article has no conflict of interest or competing interests.

Ethical approval

Not applicable.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Molina, E.D., Rodríguez, J.M., Sánchez, J.L. et al. Applications of the inverse degree index to molecular structures. J Math Chem 62, 228–249 (2024). https://doi.org/10.1007/s10910-023-01526-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-023-01526-z

Keywords

Mathematics Subject Classification

Search

Navigation