10.57647/ijm2c.2025.150425

Closed-Form Formulas for Classical and Reverse Degree-Based Topological Indices inTwo-Dimensional Grid Graphs

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  • Sara Shokrollahi Yancheshmeh*,

Received: 07-07-2025

Revised: 03-08-2025

Accepted: 12-08-2025

Published in Issue 31-08-2025

Copyright (c) 2025 Sara Shokrollahi Yancheshmeh (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Shokrollahi Yancheshmeh, S. (2025). Closed-Form Formulas for Classical and Reverse Degree-Based Topological Indices inTwo-Dimensional Grid Graphs. International Journal of Mathematical Modelling & Computations, 15(4). https://doi.org/10.57647/ijm2c.2025.150425
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Abstract

Topological indices are fundamental descriptors in mathematical chemistry, providing quantitative measures of molecular structure that correlate with physicochemical properties. This investigation presents a systematic computational approach to determineexa ct closed-form expressions for various degree-based topological indices applied
to two-dimensional grid networks Pm × Pn. We establish explicit formulas for classical Zagreb indices, Randić connectivity indices, atom-bond connectivity descriptors, geometric-arithmetic indices, harmonic indices, and the recently introduced reverse versions of these indices. Our methodology employs vertex degree distribution analysis combined with edge-based summation techniques to derive mathematically rigorous expressions. The obtained results demonstrate that grid networks exhibit predictable scaling behaviors for all examined indices, with computational complexity remaining polynomial in network dimensions. Moreover, for the first time, we provide a detailed graphical and comparative analysis of the specific degree-based topological indices addressed in this work and their reverse counterparts on grid networks, filling a gap in the literature by extending such comparisons beyond traditional molecular systems.

Keywords

  • Topological indices, Grid graphs, Degree-Based Indices, Reverse indices
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References

  1. Balaban AT. Applications of graph theory in chemistry. J Chem Inf Comput Sci. 1985;25(3):334–343. doi: 10.1021/ci00047a033
  2. Barabási A-L. Network science. Philos Trans R Soc A. 2013;371(1987):20120375. doi: 10.1098/rsta.2012.0375
  3. Erciyes K. Graph-theoretical analysis of biological net- works: a survey. Computation. 2023;11(10):188. doi: 10.3390/computation11100188
  4. Newman MEJ. Networks. 2nd ed. Oxford University Press; 2018.
  5. Trinajstić N. Chemical Graph Theory. 2nd ed. CRC Press; 1992.
  6. Aqib M, Malik MA, Afzal HU, Fatima T, Ali Y. On topological indices of some chemical graphs. Mol Phys. 2023;e2242676. doi: 10.1080/00268976.2023.2242676
  7. Estrada E. The Structure of Complex Networks: Theory and Appli- cations. Oxford University Press; 2011.
  8. Gutman I, Milojković A. GA index of molecular graphs. Croat Chem Acta. 2011;84(2):229–233. doi: 10.5562/cca1775
  9. Todeschini R, Consonni V. Handbook of Molecular Descriptors. Wiley-VCH; 2000.
  10. Ullah A, Zaman S, Hamraz A, Muzammal M. On the construction of some bioconjugate networks and their structural modeling via irregularity topological indices. Eur Phys J E. 2023;46(8):72. doi: 10.1140/epje/s10189-023-00331-5
  11. Balaban AT, Ivanciuc O. Historical development of topological in- dices. In: Devillers J, Balaban AT, editors. Topological Indices and Related Descriptors in QSAR and QSPR. CRC Press; 2000. p. 31– 68.
  12. Dearden JC. The use of topological indices in QSAR and QSPR modeling. In: Roy K, editor. Advances in QSAR Model- ing: Applications in Pharmaceutical, Chemical, Food, Agricul- tural and Environmental Sciences. Springer; 2017. p. 57–88. doi: 10.1007/978-3-319-56850-8_2
  13. Gutman I. Degree-based topological indices. Croat Chem Acta. 2013;86(4):351–361. doi: 10.5562/cca2294
  14. Hasani M, Ghods M. Topological indices and QSPR analysis of some chemical structures applied for the treatment of heart patients. Int J Quantum Chem. 2024;124(1):e27234. doi: 10.1002/qua.27234
  15. Yaghi OM, O’Keeffe M, Ockwig NW, Chae HK, Eddaoudi M, Kim J. Reticular synthesis and the design of new materials. Nature. 2003;423(6941):705–714. doi: 10.1038/nature01650
  16. Zhou HC, Kitagawa S. Metal-organic frameworks (MOFs). Chem Soc Rev. 2014;43(16):5415–5418. doi: 10.1039/C4CS90059F
  17. Delgado Friedrichs O, Dress AWM, Huson DH, Klinowski J, Mackay AL. Systematic enumeration of crystalline networks. Nature. 1999;400(6745):644–647. doi: 10.1038/23210
  18. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric field effect in atomically thin carbon films. Science. 2004;306(5696):666–669. doi: 10.1126/science.1102896
  19. Ahmad A, Koam AN, Azeem M. Reverse-degree-based topological indices of fullerene cage networks. Mol Phys. 2023;121(14):e2212533. doi: 10.1080/00268976.2023.2212533
  20. Hakeem A, Katbar NM, Shaikh H, Tolasa FT, Abro OA. Reverse degree-based topological indices study of molecular structure in tri- angular Υ-graphyne and triangular Υ-graphyne chain. Front Phys. 2024;12:1422098. doi: 10.3389/fphy.2024.1422098
  21. Khabyah A, Ahmad A, Azeem M, Ahmad Y, Koam AN. Reverse-degree-based topological indices of two-dimensional coronene fractal structures. Phys Scr. 2023;99(1):015216. doi: 10.1088/1402-4896/ad0c9c
  22. Koam AN, Ahmad A, Ahmad Y. Computation of reverse degree- based topological indices of hex-derived networks. AIMS Math. 2021;6(10):11330–11345. doi: 10.3934/math.2021656
  23. Kulli VR. Reverse Zagreb and reverse hyper-Zagreb indices of cer- tain nanostructures. Ann Pure Appl Math. 2018;18(1):47–51. doi: 10.22457/apam.v18n1a6
  24. Gutman I, Trinajstić N. Graph theory and molecular orbitals. To- tal ????-electron energy of alternant hydrocarbons. Chem Phys Lett. 1972;17(4):535–538. doi: 10.1016/0009-2614(72)85099-1
  25. Nikolić S, Kovačević G, Miličević A, Trinajstić N. The Zagreb in- dices 30 years after. Croat Chem Acta. 2003;76(2):113–124.
  26. Zhou B, Trinajstić N. The Zagreb indices: A brief review. MATCH Commun Math Comput Chem. 2004;50(2):193–202.
  27. Randić M. Characterization of molecular branching. J Am Chem Soc. 1975;97(23):6609–6615. doi: 10.1021/ja00856a001
  28. Estrada E, Torres L, Rodríguez L, Gutman I. An atom-bond connec- tivity index: modelling the enthalpy of formation of alkanes. Indian J Chem, Sect A. 1998;37:849–855.
  29. Acharya BD, Gill MK. On the index of gracefulness of a graph and the gracefulness of two-dimensional square lattice graphs. Indian J Math. 1981;23:81–94.
  30. Vukičević D, Furtula B. Topological index based on the ra- tios of geometrical and arithmetical means of end-vertex de- grees of edges. J Math Chem. 2009;46(4):1369–1376. doi: 10.1007/s10910-009-9520-x