Published in Volume XXXV, Issue 1, 2025, pages 53-85, doi: 10.47743/SACS.2025.1.53

Author: U. V. C. Kumar, H. M. Nagesh, M. B. M. Reddy

Abstract

This study presents the precise formulation of the Nirmala index and the first and second inverse Nirmala indices for the double graph of the silicon carbide network Si₂C₃-I[k, l], using its M-polynomial. Entropy measures based on these indices are derived using Shannon’s entropy model. Numerical and graphical comparisons are conducted between the Nirmala indices and their corresponding entropy measures. Curve fitting and Pearson correlation analysis are used to examine their relationships. Finally, the computed indices are employed to develop a predictive model for the graph energy of Si₂C₃-I[k, l] and estimate the HOMO-LUMO gap.

Full Text (PDF)

References

[1] Deepa Balasubramaniyan, Natarajan Chidambaram, Vignesh Ravi, and Muhammad Kamran Siddiqui. QSPR analysis of anti-asthmatic drugs using some new distance-based topological indices: A comparative study. International Journal of Quantum Chemistry, 124(9):e27372, 2024. doi:10.1002/qua.27372.

[2] Zengqiang Chen, Matthias Dehmer, and Yongtang Shi. A note on distance-based graph entropies. Entropy, 16(10):5416–5427, 2014. doi:10.3390/e16105416.

[3] Shibsankar Das and Virendra Kumar. Investigation of closed derivation formulas for GQ and QG indices of a graph via M-polynomial. Iranian Journal of Mathematical Chemistry, 13(2):129–144, 2022. doi:10.22052/ijmc.2022.246172.1614.

[4] Shibsankar Das and Virendra Kumar. On M-polynomial of the two-dimensional silicon-carbons. Palestine Journal of Mathematics, 11:136–157, 2022.

[5] Shibsankar Das and Virendra Kumar. Neighborhood degree sum-based molecular indices and their comparative analysis of some silicon carbide networks. Physica Scripta, 99:055941, 2024. doi:10.1088/1402-4896/ad3682.

[6] Shibsankar Das, Virendra Kumar, and Jayjit Barman. Geometric- quadratic and quadratic-geometric indices-based entropy measures of silicon carbide networks. Silicon, 17:75–91, 2025. doi:10.1007/s12633-024-03173-8.

[7] Shibsankar Das and Shikha Rai. M-polynomial and related degree- based topological indices of the third type of chain Hex-derived network. Malaya Journal of Matematik, 8:1842–1850, 2020. doi:10.26637/MJM0804/0085.

[8] Shibsankar Das and Shikha Rai. M-polynomial and related degree- based topological indices of the third type of Hex-derived network. Nanosystems Physics Chemistry Mathematics, 11:267–274, 2020. doi:10.17586/2220-8054-2020-11-3-267-274.

[9] Shibsankar Das and Shikha Rai. On closed derivation formulas of the Nirmala indices from the M-polynomial of a graph. Journal of the Indian Chemical Society, 100(6):101017, 2023. doi:10.1016/j.jics.2023.101017.

[10] Hanyuan Deng, Jianguang Yang, and Fangli Xia. A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes. Computers & Mathematics with Applications, 61(10):3017–3023, 2011. doi:10.1016/j.camwa.2011.03.089.

[11] Emeric Deutsch and Sandi Klavzar. M-polynomial and degree-based topological indices. Iranian Journal of Mathematical Chemistry, 6(2):93–102, 2015. doi:10.22052/ijmc.2015.10106.

[12] Boris Furtula and Mert Oz. On properties of the first inverse Nirmala index. Journal of Mathematical Chemistry, 63:96–104, 2024. doi:10.1007/s10910-024-01665-x.

[13] Ivan Gutman. Degree-based topological indices. Croatica Chemica Acta, 86:351–361, 2013. doi:10.5562/cca2294.

[14] Ivan Gutman and Nenad Trinajstic. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4):535–538, 1972. doi:10.1016/0009-2614(72)85099-1.

[15] Haruo Hosoya. On some counting polynomials in chemistry. Discrete Applied Mathematics, 19(1):239–257, 1988. doi:10.1016/0166-218X(88)90017-0.

[16] Rongbing Huang, Muhammad Siddiqui, Shazia Manzoor, Sadia Khalid, and Sultan Alaybani. On physical analysis of topological indices via curve fitting for natural polymer of cellulose network. The European Physical Journal Plus, 137, 2022. doi:10.1140/epjp/s13360-022-02629-3.

[17] Faiza Ishfaq and M. Faisal Nadeem. Topological properties of fractals via m-polynomial. Arabian Journal of Mathematics, 13:303–317, 2024. doi:10.1007/s40065-024-00466-z.

[18] Abdul Khan, Arooj Zia, Francis Campena, Muhammad Siddiqui, Fairouz Tchier, and Shahid Hussain. Investigations of entropy double & strong double graph of silicon carbide. Silicon, 16:1–11, 2024. doi:10.1007/s12633-024-02975-0.

[19] Veerabhadrappa R. Kulli. Nirmala index. International Journal of Mathematics Trends and Technology, 67(3):8–12, 2021. doi:10.14445/22315373/IJMTT-V67I3P502.

[20] Veerabhadrappa R. Kulli, Lokesha Veera, and K. Nirupadi. Computation of inverse nirmala indices of certain nanostructures. International Journal of Mathematical Combinatorics, 2:33–40, 2021.

[21] Virendra Kumar and Shibsankar Das. On Nirmala indices-based entropy measures of silicon carbide networl. Iranian Journal of Mathematical Chemistry, 14(4):271–288, 2023. doi:10.22052/ijmc.2023.252742.1704.

[22] Virendra Kumar and Shibsankar Das. Comparative study between Nirmala and Sombor indices based on their applicability, degeneracy and smoothness. The European Physical Journal Plus, 139:995, 2024. doi:10.1140/epjp/s13360-024-05768-x.

[23] Virendra Kumar and Shibsankar Das. On structure sensitivity and chemical applicability of some novel degree-based topological indices. MATCH Communications in Mathematical and in Computer Chemistry, 92:165–203, 2024. doi:10.46793/match.92-1.165K.

[24] Hadonahalli Mudalagiraiah Nagesh. QSPR analysis with curvilinear regression modeling and temperature-based topological indices, 2024. arXiv:2404.08650.

[25] Hadonahalli Mudalagiraiah Nagesh. Reverse degree-based topological indices and QSPR analysis of cancer drugs, 2024. arXiv:2403.19686.

[26] Hadonahalli Mudalagiraiah Nagesh. Topological analysis of entropy measure using regression model for terpyridine complex nanosheet, 2024. arXiv:2405.20767.

[27] Hadonahalli Mudalagiraiah Nagesh. Topological analysis of entropy measure using regression models for diamond structure, 2024. arXiv:2405.10328.

[28] Hadonahalli Mudalagiraiah Nagesh. Degree-based hybrid topological indices and entropies of wavy zigzag nanoribbons. International Journal of Quantum Chemistry, 125(7):e70028, 2025. doi:10.1002/qua.70028.

[29] Hadonahalli Mudalagiraiah Nagesh. Degree-based topological indices and QSPR analysis of Cytomegalovirus drugs. International Journal of Mathematics for Industry, 0(0):2450030, 2025. doi:10.1142/S2661335224500308.

[30] Hadonahalli Mudalagiraiah Nagesh and Muguru Chinnaswamy Mahesh Kumar. On the M-polynomial and degree-based topological indices of Dandelion graph. International Journal of Mathematical Combinatorics, 1:39–49, 2024.

[31] Hadonahalli Mudalagiraiah Nagesh, U. Vijaya Chandra Kumar, and Muhammad Kamran Siddiqui. Topological indices and entropy measures in cove-edged graphene nanoribbons using curve-fitting models. Applied Physics A, 131, 2024. doi:10.1007/s00339-024-08081-7.

[32] Hadonahalli Mudalagiraiah Nagesh and Muhammad Kamran Siddiqui. On Nirmala indices based entropy measures for the complex structure of ruthenium bipyridine, 2024. arXiv:2405.18441.

[33] Hadonahalli Mudalagiraiah Nagesh, Muhammad Kamran Siddiqui, and Imran Cheema. Topological analysis of entropy measure using regression model for silicon carbide network. The European Physical Journal Plus, 139, 2024. doi:10.1140/epjp/s13360-024-05641-x.

[34] Shikha Rai and Shibsankar Das. Comparative analysis of M-polynomial based topological indices between poly Hex-derived networks and its subdivision. Proyecciones (Antofagasta), 43(1):69–102, 2024. doi:10.22199/issn.0717-6279-5843.

[35] Milan Randic. Characterization of molecular branching. Journal of the American Chemical Society, 97(23):6609–6615, 1975. doi:10.1021/ja00856a001.

[36] Izudin Redzepovic, Boris Furtula, and Ivan Gutman. Relating total π-electron energy of benzenoid hydrocarbons with HOMO and LOMO energies. MATCH Communications in Mathematical and in Computer Chemistry, 84:229–237, 2020.

[37] Muhammad Sardar, Muhammad Asad Ali, Faraha Ashraf, and Murat Cancan. On topological indices of double and strong double graph of silicon carbide Si2C3-I[k,l]. Eurasian Chemical Communications, 5(1):37–49, 2023. doi:10.22034/ecc.2023.356160.1519. doi:10.22034/ecc.2023.356160.1519.

[38] Claude Elwood Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379–423, 1948. doi:10.1002/j.1538-7305.1948.tb01338.x.

[39] B. Gautham Shenoy, Vishu Kumar M., Divya Acharya, Hadonahalli Mudalagiraiah Nagesh, and Prasanna Poojary. Statistical analysis of Revan topological indices for drug compounds used in treatment of tuberculosis. Physica Scripta, 100, 2025. doi:10.1088/1402-4896/ada120.

[40] Haraloor Channakrishnappa Shilpa, Kamanna Gayathri, Ballenahalli Ningegowda Dharmendra, Hadonahalli Mudalagiraiah Nagesh, and Muhammad Kamran Siddiqui. On Nirmala indices–based entropy measures of silicon carbide network Si2C3-III[α,β]. Iranian Journal of Mathematical Chemistry, 2023. doi:10.22052/IJMC.2023.252742.1704.

[41] Haraloor Channakrishnappa Shilpa, Kamanna Gayathri, and Hadonahalli Mudalagiraiah Nagesh. M-polynomial and neighborhood M-polynomial of some concise drug structures: A-315675, Oseltamivir carboxylate, Zanamivir, and BCX-1812. Letters in Applied NanoBioScience, 13(2), 2024. doi:10.33263/LIANBS132.075.

[42] Haraloor Channakrishnappa Shilpa, Kamanna Gayathri, Hadonahalli Mudalagiraiah Nagesh, and Nittur Narahari. M-polynomials and degree-based topological indices of Mycielskian of paths and cycles. Communications in Mathematics and Applications, 14(4):1375–1383, 2023. doi:10.26713/cma.v14i4.2574.

[43] Haraloor Channakrishnappa Shilpa, Kamanna Gayathri, Hadonahalli Mudalagiraiah Nagesh, and Nittur Narahari. On the M-polynomials and degree-based topological indices of an important class of graphs. International Journal of Mathematical Combinatorics, 2:38–50, 2023.

[44] Nenad Trinajstic. Chemical Graph Theory. CRC Press, 2nd edition, 2018. doi:10.1201/9781315139111.

[45] Ashish Verma, Sourav Mondal, Nilanjan De, Nilanjan De, and Anita Pal. Topological properties of Bismuth tri-iodide using neighborhood M-polynomial. International Journal of Mathematics Trends and Technology, 65(10):83–90, 2019. doi:10.14445/22315373/IJMTT-V65I10P512.

[46] Xing-Long Wang, Muhammad Kamran Siddiqui, Syed Ajaz K. Kirmani, Shazia Manzoor, Sarfraz Ahmad, and Mlamuli Dhlamini. On topological analysis of entropy measures for silicon carbides networks. Complexity, 2021(1):4178503, 2021. doi:10.1155/2021/4178503.

[47] Harry Wiener. Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1):17–20, 1947. doi:10.1021/ja01193a005.

[48] Jun Yang, Muhammad Kamran Siddiqui, Mazhar Hussain, Shazia Manzoor, Nazir Hussain, and Zohaib Saddique and. QSPR analysis of some phenolic compounds present in Moringa Oleifera. Polycyclic Aromatic Compounds, 44(6):4197–4214, 2024. doi:10.1080/10406638.2023.2247120.

[49] Xiujun Zhang, S. Prabhu, M. Arulperumjothi, S. Manikanda Prabhu, Micheal Arockiaraj, and V. Manimozhi. Distance based topological characterization, graph energy prediction, and NMR patterns of benzene ring embedded in P-type surface in 2D network. Scientific Reports, 14:23766, 2024. doi:10.1038/s41598-024-75193-8.

Bibtex

@article{sacscuza:nagesh2025dtcgepscndgo, 
	    title={Degree-based Topological Characterization and Graph Energy Prediction of Silicon Carbide Network $Si_2C_3-I[k, l]$ via Double Graph Operator}, 
	    author={U. V. C. Kumar, H. M. Nagesh, M. B. M. Reddy},
	    journal={Scientific Annals of Computer Science}, 
	    volume={35},
	    number={1}, 
	    organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania}, 
	    year={2025}, 
	    pages={53-85},
	    publisher={Alexandru Ioan Cuza University Press, Ia\c si},
	    doi={10.47743/SACS.2025.1.53} 
	    }