Expected Values of Molecular Descriptors in Random Polyphenyl Chains

Zahid Raza, Kiran Naz, Sarfraz Ahmad


A chemical graph is a model used to indicate a chemical combination. In a molecular graph, vertices define atoms, and edges are represented as chemical bonds. A topological index is a single number to characterize the graph of a molecule. In this article, we study the topological properties of some special chains. The polyphenyl chains with hexagons are graphs of aromatic organic compounds. The key purpose of this article is to explore the expected value of Sombor, reduced Sombor, and average Sombor index for this category of organic compounds. It was investigated that the Sombor, reduced Sombor and average Sombor index revealed adequate discriminative potential of alkanes. It has been tested that these indicators can be used effectively in modeling chemical thermodynamic structures. The average value of the Sombor, reduced Sombor, and average Sombor index for the set of all spiro and random polyphenyl chains has been determined. Finally, the ratio between the expected values of these mentioned indices for both random chains has been resolved.


Doi: 10.28991/ESJ-2022-06-01-012

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Sombor Index; Reduced Sombor Index; Average Sombor Index; Spiro Chain; Polyphenyl Chains.


Huang, G., Kuang, M., & Deng, H. (2015). The expected values of Kirchhoff indices in the random polyphenyl and spiro chains. Ars Mathematica Contemporanea, 9(2), 207–217. doi:10.26493/1855-3974.458.7b0.

Fang, X., You, L., & Liu, H. (2021). The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs. International Journal of Quantum Chemistry, 121(17). doi:10.1002/qua.26740.

Reti, T., Dosli, T., & Ali, A. (2021). On the Sombor index of graphs. Contributions to Mathematics, 3, 11–18. doi:10.47443/cm.2021.0006.

Ulker, A., Gursoy, A., Gursoy, N. K., & Gutman, I. (2021). Relating graph energy and Sombor index. In Discrete Mathematics Letters (Vol. 8, pp. 6–9). doi:10.47443/dml.2021.0085.

Bonchev, D. (1983). Information theoretic indices for characterization of chemical structures (No. 5). Research Studies Press.

Deng, H. (2012). Wiener indices of spiro and polyphenyl hexagonal chains. Mathematical and Computer Modelling, 55(3–4), 634–644. doi:10.1016/j.mcm.2011.08.037.

Wu, T., & Lü, H. (2019). Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. In Open Mathematics, 17(1), , 668–676. doi:10.1515/math-2019-0053.

Redžepović, I. (2021). Chemical applicability of Sombor indices. Journal of the Serbian Chemical Society, 86(5), 445–457. doi:10.2298/JSC201215006R.

Deng, H., & Tang, Z. (2014). Kirchhoff indices of spiro and polyphenyl hexagonal chains. Utilitas Mathematica, 95, 113-128.

Došlic, T., & Litz, M. S. (2012). Matchings and independent sets in polyphenylene chains. Match-Communications in Mathematical and Computer Chemistry, 67(2), 313.

Yang, Y., Liu, H., Wang, H., & Sun, S. (2017). On Spiro and polyphenyl hexagonal chains with respect to the number of BC-subtrees. International Journal of Computer Mathematics, 94(4), 774–799. doi:10.1080/00207160.2016.1148811.

Raza, Z. (2020). The Harmonic and Second Zagreb Indices in Random Polyphenyl and Spiro Chains. Polycyclic Aromatic Compounds. doi:10.1080/10406638.2020.1749089.

Wang, G., Li, X., & Li, C. (2014). Polyphenyl Systems with Extremal Hyper-Wiener Indices. Journal of Computational and Theoretical Nanoscience, 11(4), 1129-1132. https://doi.org/10.1166/jctn.2014.3472.

Raza, Z. (2020). The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains. Heliyon, 6(7). doi:10.1016/j.heliyon.2020.e04479.

Bai, Y., Zhao, B., & Zhao, P. (2009). Extremal merrifield-simmons index and Hosoya index of polyphenyl chains. Match, 62(3), 649–656.

Chen, A., & Zhang, F. (2009). Wiener index and perfect matchings in random phenylene chains. Match, 61(3), 623–630.

Raza, Z.; Imran, M. (2021). Expected Values of Some Molecular Descriptors in Random Cyclooctane Chains. Symmetry, 13(11), 2197. doi:10.3390/sym13112197.

Bian, H., & Zhang, F. (2009). Tree-like polyphenyl systems with extremal Wiener indices. Match, 61(3), 631–642.

Bureš, M., Pekárek, V., & Ocelka, T. (2008). Thermochemical properties and relative stability of polychlorinated biphenyls. Environmental Toxicology and Pharmacology, 25(2), 148–155. doi:10.1016/j.etap.2007.10.010.

Yang, Y., Liu, H., Wang, H., & Fu, H. (2015). Subtrees of spiro and polyphenyl hexagonal chains. Applied Mathematics and Computation, 268, 547–560. doi:10.1016/j.amc.2015.06.094.

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DOI: 10.28991/ESJ-2022-06-01-012


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