New Paradigmatic Framework of Entropy Computing in Benzenoids Octahedral Structures
DOI:
https://doi.org/10.11113/mjfas.v22n3.5139Abstract
Network entropy is a potent quantitative approach to the analysis of the structural complexity of connected systems, derived out of information theory and has received significant interest in computer science, the biological sciences, and molecular chemistry because of its capacity to reflect high structural diversity and low symmetry. Topological indices give a graph entropy measurement which is also an efficient and systematic set of indices to study the structure of molecules and predict their behaviour. This work is a research on the K-Banhatti entropy of three benzenoid octahedral structures, i.e. BPOS, BDPOS, and BHPOS, and the purpose of this investigation is to use the data on the entropy to point out the structural features of these structures. Topological descriptors of molecular structures are an important part of the chemical graph theory and are widely used in cheminformatics, information technology, biological research, and other areas. There has also been a broad selection of topological indices designed to represent the behavior of molecules especially non-regular graphs where the irregularity measures are able to provide a better insight into the structural heterogeneity. Inevitably, we analyse a few indices of irregularity in order to describe the non-uniformity of benzenoid networks and aid quantitative structure-activity relationship (QSAR) analysis. A more detailed insight into structural organization of benzenoid octahedral structures is made possible through combined application of entropy measures and irregularity indices. Further, we compute the first and second kinds of modified Zagreb entropies for BPOS, BDPOS, and BHPOS structures. The findings reveal that the suggested descriptors are robust, informative and computationally efficient measures used in examining complex molecular networks and thus, serve in the development of entropy-based measures in chemical graph theory.
References
Rada, Juan, Roberto Cruz, and Ivan Gutman. ”Benzenoid systems with extremal vertex-degree-based topological indices.” MATCH Commun. Math. Comput. Chem 72.1 (2014): 125-136. https://doi.org/10.47000/tjmcs.886707
Akhter, Shehnaz, and Muhammad Imran. ”On molecular topological properties of benzenoid struc tures.”
Canadian Journal of Chemistry 94.8 (2016): 687-698. https://doi.org/10.1139/cjc-2016-0032
Ivanciuc, Ovidiu. ”QSAR comparative study of Wiener descriptors for weighted molecular graphs.” Journal of chemical information and computer sciences 40.6 (2000): 1412-1422. https://doi.org/10.1021/ci000068y
Gowtham, K. J.; Husin, M. N.; A Study of Families of Bistar and Corona Product of Graph: Reverse Topological Indices. Malaysian Journal of Mathematical Sciences, 2023, 17(4), 575-586. https://doi.org/10.47836/mjms.17.4.04
Gowtham, K. J.; Husin, M. N.; Siddiqui, M. K.; Some New Bounds on the Modified Symmetric Division Deg Index. Malaysian Journal of Mathematical Sciences, 2024, 18(1), 39-50. https://doi.org/10.47836/mjms.18.1.03
Rafiullah, M.; Dur-E-Jabeen; Husin, M. N.; Some Mathematical Properties of Sombor Indices for Regular Graphs, Malaysian Journal of Fundamental and Applied Sciences, 2024, 20(6), 1392-1397. https://doi.org/10.11113/mjfas.v20n6.3839
Rosary, Maria Singaraj, et al. ”A Perspective Approach to Study the Valency-Based Irregular Indices for BPOSs.” Journal of Chemistry 2023 (2023). https://doi.org/10.1155/2023/5521743
Liu, Jia-Bao, et al. ”On topological properties for BPOS networks.” Molecules 27.19 (2022): 6366.
https://doi.org/10.3390/molecules27196366
Dustigeer, Ghulam, et al. ”On multiplicative degree based topological indices for planar octahedron networks.”
Main Group Metal Chemistry 43.1 (2020): 219-228. https://doi.org/10.1515/mgmc-2020-0026
Chu, Zheng-Qing, et al. ”Some topological indices of dendrimers determined by their Banhatti poly nomials.”
Heterocyclic Communications 26.1 (2020): 99-111. https://doi.org/10.1515/hc-2020-0012
Saleem, H.; Husin, M. N.; Ali, S.; Hameed, M. S.; Ahmad, Z. Topology of Edge Contracted Möbius Ladder: Indices and Dimension. Malaysian Journal of Fundamental and Applied Sciences. 2024, 20(4), 739-758. https://doi.org/10.11113/mjfas.v20n4.3386
Fathi, A.; Sahaya, V. J.; Husin, M. N.; Augustine, T.; Valency-Based Molecular Descriptor on Structural Property Relationship of Ni Tetrathiafulvalene Tetrathionate. Malaysian Journal of Fundamental and Applied Sciences. 2024, 20(6), 1398-1409. https://doi.org/10.11113/mjfas.v20n6.3831
Husin, M. N.; Saudi, N. H. A. M.; Investigation of Zagreb indices and Zagreb coindices of line graphs implementing subdivision process, AIP Conference Proceeding., 2024, 2905(1), 030002. https://doi.org/10.1063/5.0172141
Dustigeer, Ghulam, et al. ”On multiplicative degree based topological indices for planar octahedron networks.”
Main Group Metal Chemistry 43.1 (2020): 219-228. https://doi.org/10.1515/mgmc-2020-0026
Ali, Ashaq, et al. ”M-polynomials and topological indices of zigzag and rhombic benzenoid systems.” Open chemistry 16.1 (2018): 73-78. https://doi.org/10.1515/chem-2018-0008
Rada, Juan, Roberto Cruz, and Ivan Gutman. ”Benzenoid systems with extremal vertex-degree-based topological indices.” MATCH Commun. Math. Comput. Chem 72.1 (2014): 125-136. https://doi.org/10.47000/tjmcs.886707
Akhter, Shehnaz, and Muhammad Imran. ”On molecular topological properties of benzenoid struc tures.” Canadian Journal of Chemistry 94.8 (2016): 687-698.https://doi.org/10.1139/cjc-2016-0032
Mueller, Wolfgang R., et al. ”Molecular topological index.” Journal of chemical information and computer sciences 30.2 (1990): 160-163. https://doi.org/10.1021/ci00066a011
Babujee, J. Baskar, and S. Ramakrishnan. ”Topological indices and new graph structures.” Applied Mathematical Sciences 6.108 (2012): 5383-5401. https://doi.org/10.12988/ams.2012.12462
Ghani, Muhammad Usman, et al. ”Entropies Via Various Molecular Descriptors of Layer Structure of H 3 BO
” Mathematics 10.24 (2022): 4831. https://doi.org/10.3390/math10244831
Bromiley, P. A., N. A. Thacker, and E. Bouhova-Thacker. ”Shannon entropy, Renyi entropy, and information.”
Statistics and Inf. Series (2004-004) 9 (2004): 2-8. https://doi.org/10.1117/12.533256
Ali, Shahbaz, et al. ”On rotationally symmetrical planar networks and their local fractional metric Dimension.”
Symmetry 15.2 (2023): 530. https://doi.org/10.3390/sym15020530
Tag El Din, El Sayed M., et al. ”Some Novel Results Involving Prototypical Computation of Zagreb Polynomials and Indices for SiO 4 Embedded in a Chain of Silicates.” Molecules 28.1 (2022): 201. https://doi.org/10.3390/molecules28010201
Qasim, Muhammad, Hani Shaker, and Mian Muhammad Zobair. ”Physical Correlation of Topolog ical Indices of Transition Metal Tetra-Cyano Benzene Structure Using Curve Fitting.” Polycyclic Aromatic Compounds 43.8 (2023): 7518-7530. https://doi.org/10.1080/10406638.2022.2144885
Salman, Muhammad, et al. ”Some Valency Oriented Molecular Invariants of Certain Networks.” Combinatorial Chemistry & High Throughput Screening 25.3 (2022): 462-475. https://doi.org/10.2174/1386207324666210304101153
Almalki, Norah, and Hafsah Tabassum. ”On K-Banhatti, Revan Indices and Entropy Measures of MgO (111)
Nanosheets via Linear Regression.” Mathematics 12.4 (2024): 561. https://doi.org/10.3390/math12040561
Alali, A. S., et al. ”Algebraic structure graphs over the commutative ring Zm: Exploring topological indices and entropies using M-polynomials, Mathematics, 11 (2023), 3833.” https://doi.org/10.3390/math11183833
Fathi, A.; Augustine, T.; Husin, M. N.; Vijay, J. S.; Thomas, J. M.; Roy, .; Graph Theoretical Approach to Topological Features of Zeolitic Tetrahedral Imidazolate Framework-3, Malaysian Journal of Fundamental and Applied Sciences, 2025, 21(2), 1823-1831. https://doi.org/10.11113/mjfas.v21n2.4127
Bayati, J. H. H.; Al-Taai, E. A.; Husin, M. N.; Shahid, T.; Waqas, M.; Muda, Y.; A Study of Novel Molecular Descriptors with QSPR analysis of Lung Cancer Drugs, Communications in Mathematical Biology and Neuroscience, 2025, 148. https://doi.org/10.28919/cmbn/8412
Vijay, S. J.; Roy, S.; Ponnusamy, C. S.; Thomas, J. M.; Husin, M. N.; Fathi, A.; Efficient Generation of Specific Counting Polynomials and their Subsequent Indices of Complex Boric Acid Structure, Current Organic Synthesis, 2025, 22(7), 821-828. https://doi.org/10.2174/157017942266624021511342
Vijay, J. S.; Roy, S.; Azeem, M.; Augustine, T.; Husin, M. N.; Quantitative Structure-Property Relationship (QSPR) Modeling of Central Nervous System (CNS) Drug Activity using Molecular Descriptors, Current Organic Synthesis, 2025, 22(7), 799-810. https://doi.org/10.2174/157017942266624010815410
Renai, P. N. A. D.; Roy, S.; Husin, M. N.; On Shannon’s Strategy of Computing Degree Based Entropy Measures for Melamine Cyanuric Acid Molecular Structure, Malaysian Journal of Mathematical Sciences, 2025, 19(1), 241-267. https://doi.org/10.47836/mjms.19.1.12
Jia-Bao Liu, Xue Zhang, and Jinde Cao. Structural properties of extended pseudo-fractal scale-free network with higher network efficiency, Journal of Complex Networks. 2024, 12(3): cnae023. https://doi.org/10.1093/comnet/cnae023
Qummer, A. A., Saqib, M., Ali, S., Ashebo, M. A., & Jumaniyazov, S. (2026). A novel computational study of reverse degree topological indices and entropies for nanostar dendrimers. Scientific Reports. https://doi.org/10.1038/s41598-026-11492-x
Mustafa, G., Shoaib, M., Ali, S., & Husin, M. N. (2025). On Characterizations of Molecular Descriptors and Entropies for Oxytocin and Cholic Acid Chemical Networks. Malaysian Journal of Fundamental and Applied Sciences, 21(5), 2697-2714. https://doi.org/10.11113/mjfas.v21n5.4423
Ali, S., Husin, M. N., Akram, T., & Hameed, M. S. (2025). On topological characterizations and entropies for algebraic structure graphs over Dihedral groups. Discrete Mathematics, Algorithms and Applications, 2550155. https://doi.org/10.1142/S179383092550155X
Ali, S., Ashraf, S., Ali, S., Afzal, A., & Alali, A. S. (2024). On Local Fractional Topological Indices and Entropies for Hyper-Chordal Ring Networks Using Local Fractional Metric Dimension. Symmetry, 17(1), 5. https://doi.org/10.3390/sym17010005
Ali, S., Shang, Y., Hassan, N., & S. Alali, A. (2024). On topological indices and entropy dynamics over zero divisors graphs under cartesian product of commutative rings. Research in Mathematics, 11(1), 2427339. https://doi.org/10.1080/27684830.2024.2427339
Ali, S., Mahmood, M. K., & Ghaffar, S. (2024). Sev eral topological indices and entropies for certain families of commutative graphs over Quaternion groups. VFAST Transactions on Mathematics, 12(2), 32-48. https://doi.org/10.21015/vtm.v12i2.1742
Ghani, M. U., Ali, S., Imran, M., Karamti, H., Sultan, F., & Almusawa, M. Y. (2023). Hex-derived molecular descriptors via generalized valency-based entropies. IEEE Access, 11, 42052-42068. https://doi.org/10.1109/ACCESS.2023.3268412
Chen, Z.; Dehmer, M.; Shi, Y. A note on distance-based graph entropies. Entropy 2014, 16, 5416–5427. https://doi.org/10.3390/e16105416
Manzoor, S.; Siddiqui, M.K.; Ahmad, S. On entropy measures of molecular graphs using topological indices. Arab. J. Chem. 2020, 13, 6285–6298. https://doi.org/10.1016/j.arabjc.2020.05.019
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Ashfaq Ahmed Qummer, Muhammad Saqib, Mohamad Nazri Husin, Shahbaz Ali, Rahmawati Rahmawati

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














