On ‎Nirmala ‎Indices-based ‎Entropy Measures of ‎Silicon ‎Carbide Network

Document Type : Research Paper


Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, Uttar Pradesh, India.


‎Topological indices are numerical parameters for understanding the fundamental topology of chemical structures that correlate with the quantitative structure-property relationship (QSPR)‎ / ‎quantitative structure-activity relationship (QSAR) of chemical compounds‎. ‎The M-polynomial is a modern mathematical approach to finding the degree-based topological indices of molecular graphs‎.
‎Several graph assets have been employed to discriminate the construction of entropy measures from the molecular graph of a chemical compound‎. ‎Graph entropies have evolved as information-theoretic tools to investigate the structural information of a molecular graph‎. ‎The possible applications of graph entropy measures in chemistry‎, ‎biology and discrete mathematics have drawn the attention of researchers‎. ‎In this research work‎, ‎we compute the Nirmala index‎, ‎first and second inverse Nirmala index for silicon carbide network $Si_{2}C_{3}\textit{-I}[p,q]$ with the help of its M-polynomial‎. ‎Further‎, ‎we introduce the concept of Nirmala indices-based entropy measure and enumerate them for the above-said network‎. ‎Additionally‎, ‎the comparison and correlation between the Nirmala indices and their associated entropy measures are presented through numerical computation and graphical approaches‎. ‎Following that‎, ‎curve fitting and correlation analysis are performed to investigate the relationship between the Nirmala indices and corresponding entropy measures.


Main Subjects

[1] D. B. West, Introduction to Graph Theory, Prentice Hall, 2nd edition, 2000.
[2] H. Wagner and H. Wang, Introduction to Chemical Graph Theory, Discrete Mathematics and its Applications, CRC Press, 2019.
[3] N. Trinajstic, Chemical Graph Theory, CRC press, 2018.
[4] S. Das, S. Rai and V. Kumar, On topological indices of Molnupiravir and its QSPR modelling with some other antiviral drugs to treat COVID-19 patients, J. Math. Chem. (2023) 1–44, https://doi.org/10.1007/s10910-023-01518-z.
[5] H. Deng, J. Yang and F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comput. Math. with Appl. 61 (2011) 3017– 3023, https://doi.org/10.1016/j.camwa.2011.03.089.
[6] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20, https://doi.org/10.1021/ja01193a005.
[7] M. Randic, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609– 6615, http://dx.doi.org/10.1021/ja00856a001.
[8] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538, https://doi.org/10.1016/0009-2614(72)85099-1.
[9] S. Das, S. Rai and M. Mandal, M-polynomial and correlated topological indices of antiviral drug Molnupiravir used as a therapy for COVID-19, Polycycl. Aromat. Compd. 43 (2023) 7027–7041, https://doi.org/10.1080/10406638.2022.2131854.
[10] D. Balasubramaniyan and N. Chidambaram, On some neighbourhood degree-based topological indices with QSPR analysis of asthma drugs, Eur. Phys. J. Plus 138 (2023) p. 823, https://doi.org/10.1140/epjp/s13360-023-04439-7.
[11] A. Rauf, M. Naeem and A. Hanif, Quantitative structure–properties relationship analysis of eigen-value-based indices using COVID-19 drugs structure, Int. J. Quantum Chem. 123 (2023) p. e27030, https://doi.org/10.1002/qua.27030.
[12] V. R. Kulli, Nirmala index, Int. J. Math. Trends Technol. 67 (2021) 8–12, https://doi.org/10.14445/22315373/IJMTT-V67I3P502.
[13] V. R. Kulli, V. Lokesha and K. Nirupadi, Computation of inverse Nirmala indices of certain nanostructures, International J. Math. Combin. 2 (2021) 33–40.
[14] H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988) 239–257, https://doi.org/10.1016/0166-218X(88)90017-0.
[15] A. Verma, S. Mondal, N. De and A. Pal, Topological properties of bismuth tri-iodide using neighborhood M-polynomial, Int. J. Math. Trends Technol. 67 (2019) 83–90, https://doi.org/10.14445/22315373/IJMTT-V65I10P512.
[16] E. Deutsch and S. Klavžar, M-polynomial and degree-based topological indices, Iran. J. Math. Chem., 6 (2015) 93–102, https://doi.org/10.22052/ijmc.2015.10106.
[17] Y. C. Kwun, M. Munir, W. Nazeer, S. Rafique and S. M. Kang, M-polynomials and topological indices of V-Phenylenic nanotubes and nanotori, Sci. Rep. 7 (2017) p. 8756, https://doi.org/10.1038/s41598-017-08309-y.
[18] M. Munir, W. Nazeer, S. Rafique and S. M. Kang, M-polynomial and related topological indices of nanostar dendrimers, Symmetry 8 (2016) p. 97, https://doi.org/10.3390/sym8090097.
[19] S. Das and S. Rai, M-polynomial and related degree-based topological indices of the third type of Hex-derived network, Nanosyst.: Phys. Chem. Math. 11 (2020) 267–274, https://doi.org/10.17586/2220-8054-2020-11-3-267-274.
[20] S. Das and S. Rai, M-polynomial and related degree-based topological indices of the third type of chain Hex-derived network, Malaya J. Mat. 8 (2020) 1842–1850, https://doi.org/10.26637/MJM0804/0085.
[21] S. Das and V. Kumar, On M-polynomial of the two-dimensional silicon-carbons, Palest. J. Math. 11 (2022) 136–157.
[22] S. Das and V. Kumar, Investigation of closed derivation formulas for GQ and QG indices of a graph via M-polynomial, Iran. J. Math. Chem. 13 (2022) 129–144, https://doi.org/10.22052/ijmc.2022.246172.1614.
[23] S. Das and S. Rai, On closed derivation formulas of Nirmala indices from the M-polynomial of a graph, J. Indian Chem. Soc. 100 (2023) p. 101017, https://doi.org/10.1016/j.jics.2023.101017.
[24] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379–423, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
[25] R. Huang, M. K. Siddiqui, S. Manzoor, S. Khalid and S. Almotairi, On physical analysis of topological indices via curve fitting for natural polymer of cellulose network, Eur. Phys. J. Plus 137 (2022) 1–17, https://doi.org/10.1140/epjp/s13360-022-02629-3.
[26] X. L. Wang, M. K. Siddiqui, S. A. K. Kirmani, S. Manzoor, S. Ahmad and M. Dhlamini, On topological analysis of entropy measures for silicon carbides networks, Complexity 2021 (2021) 1–26, https://doi.org/10.1155/2021/4178503.
[27] S. Manzoor, M.K. Siddiqui and S. Ahmad, On entropy measures of polycyclic hydroxychloroquine used for novel coronavirus (COVID-19) treatment, Polycycl. Aromat. Compd. 42 (2022) 2947–2969, https://doi.org/10.1080/10406638.2020.1852289.
[28] S. Manzoor, M.K. Siddiqui and S. Ahmad, On entropy measures of molecular graphs using topological indices, Arab. J. Chem. 13 (2020) 6285–6298, https://doi.org/10.1016/j.arabjc.2020.05.021.
[29] Z. Chen, M. Dehmer and Y. Shi, A note on distance-based graph entropies, Entropy 16 (2014) 5416–5427, https://doi.org/10.3390/e16105416.
[30] P. Li, R. Zhou and X. C. Zeng, The search for the most stable structures of silicon–carbon monolayer compounds, Nanoscale 6 (2014) 11685–11691, https://doi.org/10.1039/C4NR03247K.
[31] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. E. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669, https://doi.org/10.1126/science.1102896.
[32] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin and A. K. Geim, Unconventional quantum hall effect and berry’s phase of 2$\pi$ in bilayer graphene, Nature Phys. 2 (2006) 177–180, https://doi.org/10.1038/nphys245.