M-Polynomials and Degree-Based Topological Indices of the Two Dimensional Coronene Fractal Structures
Abstract
Topological indices are numerical parameters used to study the physical and chemical properties of compounds. These indices, particularly degree-based topological indices, have shown significant correlations with various properties of compounds, making them a focal point of extensive research. The M-polynomial plays a crucial role in influencing degree-based topological indices. In this study, we derive closed-form expressions for several prominent degree-based topological indices of two-dimensional coronene fractal structures. These indices include the first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index, the inverse Randić index, and the augmented Zagreb index, using calculus-based methods.
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References
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[2] Z. Ahmad, M. Naseem, M. Naseem, M. K. Jamil, M. F. Nadeem, and S. Wang, “Eccentric connectivity indices of titania nanotubes TiO2[m; n],” Eurasian Chemical Communications, vol. 2, no. 6, pp. 712–721, 2020.
[3] Z. Ahmad, M. Naseem, M. Naseem, M. Kamran Jamil, M. Kamran Siddiqui, and M. Faisal Nadeem, “New results on eccentric connectivity indices of V-Phenylenic
nanotube, ”Eurasian Chemical Communications, vol. 2, no. 6, pp. 663–671, 2020.
[4] F. Afzal, M. A. Razaq, M. Abdul Razaq, D. Afzal, and S. Hameed, “Weighted entropy of penta chains graph,” Eurasian Chemical Communications, vol. 2, no. 6, pp. 652–662, 2020.
[5] M. S. Ahmad, W. Nazeer, S. M. Kang, M. Imran, and W. Gao, “Calculating degreebased topological indices of dominating David derived networks,” Open Physics, vol.15, no. 1, pp. 1015–1021, 2017.
[6] F. Afzal, S. Hussain, D. Afzal, and S. Razaq, “Some new degree based topological indices via M-polynomial,” Journal of Information and Optimization Sciences, vol. 41,no. 4, pp. 1061–1076, 2020.
[7] A. Q. Baig, M. Naeem, M. Naeem, W. Gao, and J.-B. Liu, “General fifth M-Zagreb indices and fifth M-Zagreb polynomials of carbon graphite,” Eurasian Chemical Communications, vol. 2, no. 3, pp. 634–640, 2020.
[8] J. B. Babujee and S. Ramakrishnan, “Topological indices and new graph structures,”Applied Mathematical Sciences, vol. 6, no. 108, pp. 5383–5401, 2012.
[9] A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, “Molecular description of carbon graphite and crystal cubic carbon structures,” Canadian Journal of Chemistry, vol. 95,no. 6, p. 674, 2017
[10] F. M. Bruckler, T. Doslic, A. Graovac, and I. Gutman, “On a class of distancebased molecular structure descriptors, ”Chemical Physics Letters, vol. 503, no. 4–6,
pp. 336–338, 2011.
[11] F. A. Cotton, G. Wilkinson, C. A. Murillo, and M. Bochmann, Advanced Inorganic Chemistry, John Wiley and Sons, Hoboken, NJ, USA, 6th edition, 1999.
[12] M. Cancan, S. Ediz, H. Mutee-Ur-Rehman, and D. Afzal, “Mpolynomial and topological indices Poly (E+yleneAmido- Amine) dendrimers,” Journal of Information and Optimization Sciences, vol. 41, no. 4, pp. 1117–1131, 2020.
[13] M. Cancan, S. Ediz, S. Ediz, and M. R. Farahani, “On vedegree atom-bond connectivity, sum-connectivity, geometric-arithmetic and harmonic indices of copper oxide,” Eurasian Chemical Communications, vol. 2, no. 5, pp. 641–645, 2020.
[14] G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, “Graphs with maximum connectivity index,” Computational Biology and Chemistry, vol. 27, no. 1, pp. 85–90,
2003.
[15] H. Deng, J. Yang, and F. Xia, “A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes,” Computers and Mathematics with Applications, vol. 61, no. 10, pp. 3017–3023, 2011.
[16] E. Deutsch and S. Klavzar, “M-polynomial and degree-based topological indices,” Iranian Journal of Mathematical Chemistry, vol. 6, p. 93102, 2015
[17] A. A. Dobrynin, R. Entringer, and I. Gutman, “Wiener index of trees: theory and applications,” Acta Applicandae Mathematicae, vol. 66, no. 3, pp. 211–249, 2001.
[18] D. Dimitrov, “On structural properties of trees with minimal atom-bond connectivity index IV: solving a conjecture about the pendent paths of length three,” Applied Mathematics and Computation, vol. 313, pp. 418–430, 2017.
[19] E. A. Ekimov, V. A. Sidorov, E. D. Bauer, N. N. Melnik, N. J. Curro, and J. D.+ompson, “Superconductivity in diamond, ”Nature, vol. 428, p. 5425, 2004.
[20] M. R. Farahani, “Some connectivity indices and Zagreb index of polyhex nanotubes,”Acta Chimica Slovenica, vol. 59, pp. 779–783, 2012.
[21] W. Gao, Y. Wang, B. Basavanagoud, and M. K. Jamil, “Characteristics studies of molecular structures in drugs,” Saudi Pharmaceutical Journal, vol. 25, no. 4, pp.
580–586, 2017
[22] G. Gayathri, U. Priyanka, and U. Priyanka, “Degree based topological indices of zigzag chain,” Journal of Mathematics and Informatics, vol. 11, pp. 83–93, 2017.
[23] I. Gutman, “Some properties of the Winer polynomial, ”Graph Ieory Notes N. Y.vol. 125, pp. 13–18, 1993.
[24] I. Gutman and B. Furtula, Recent Results in the Ieory of Randic Index, University of Kragujevac, Kragujevac, Serbia, 2008
[25] W. Gao, Z. Iqbal, M. Ishaq, R. Sarfraz, M. Aamir, and A. Aslam, “On eccentricitybased topological indices study of a class of porphyrin-cored dendrimers,” Biomolecules, vol. 8, no. 3, p. 71, 2018.
[26] W. Gao, H. Wu, M. K. Siddiqui, and A. Q. Baig, “Study of biological networks using graph theory,” Saudi Journal of Biological Sciences, vol. 25, no. 6, pp. 1212–1219, 2018.
[27] Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, “On molecular graphs with smallest and greatest zeroth-order general Randic index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, p. 425, 2005.
[28] M. Imran, S. A. Bokhary, S. A. U. h. Bokhary, S. Manzoor, and M. K. Siddiqui, “Onmolecular topological descriptors of certain families of nanostar dendrimers,” Eurasian Chemical Communications, vol. 2, no. 6, pp. 680–687, 2020.
[29] O. Ivanciuc, “Chemical graphs, molecular matrices and topological indices in chemoinformatics and quantitative structure-activity relationships§,” Current Computer AidedDrug Design, vol. 9, no. 2, pp. 153–163, 2013.
[30] L. B. Kier and L. H. Hall, Molecular Connectivity in StructureActivity Analysis, Wiley, New York, NY, USA, 1986.
[31] M. Ahmad, A. Qayyum, G. Atta, S. S. Supadi, M. Saleem, U.Ali, A New Study on Degree Based Topological Indices of Harary Subdivision graphs With Application,
Int.J.Anal. (2024). 22:0.
[32] M. Ahmad, M. J. Hussain, G. Atta, S. Raza, I. Waheed, A. Qayyum, Topological Evaluation of Four Para-Line Graphs Absolute Pentacene Graphs Using Topological, International Journal of Analysis and Applications (2023), 21:0.
[33] M. Ahmad, S. Hussain, I. Zahid, U. parveen, M. Sultan and A. Qayyum, On degree based topological indices of petersen subdivision graph, European Journal of Mathematical Analysis 3 (2023)20.
[34] R. M. Kashif, M. Ahmad, A. Qayyum, S. S. Supadi, M. J. Hussain, S. Raza, On Degree-Based Topological Indices of Toeplitz Graphs, Int.J.Anal. (2023). 21:111.
[35] H. Yang and X. Zhang, “The independent domination numbers of strong product of two cycles,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 21, no. 7-8, pp. 1495–1507, 2018.
[36] B. Zhou and I. Gutman, “Relations between wiener, hyperwiener and Zagreb indices,”Chemical Physics Letters, vol. 394, no. 1–3, pp. 93–95, 2004.
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