On Closed Derivation Formulas over the M-polynomial for the Elliptic-Type Indices of a Graph

Document Type : Research Paper

Authors

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

10.22052/ijmc.2025.256243.1970

Abstract

‎A polynomial function that provides information about the molecular structure of a graph is known as the M-polynomial of a graph‎. ‎This polynomial helps us to understand the characteristics of chemical compounds and their relationships‎. ‎Very recently‎, ‎in 2024‎, ‎the elliptic Sombor~(\textit{ESO})‎, ‎reduced elliptic~(\textit{RE}) and modified reduced elliptic~($^{m}\textit{RE}$) indices of a graph were proposed and their values were calculated for some standard graphs‎, ‎jagged-rectangle benzenoid systems and polycyclic aromatic hydrocarbons‎. ‎In this work‎, ‎we establish closed derivation formulas for the above-mentioned elliptic-type indices of a graph based on its M-polynomial‎. ‎Moreover‎, ‎we enumerate the elliptic-type indices of the above family of chemical graphs.

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Main Subjects


[1] N. Trinajstic, Chemical Graph Theory, 2nd Edition, CRC Press, Boca Raton, 1992.
[2] I. Gutman, B. Furtula and M. S. Öz, Geometric approach to vertex-degree-based topological indices–Elliptic Sombor index, theory and application, Int. J. Quantum Chem. 124 (2024) #e27346, https://doi.org/10.1002/qua.27346.
[3] V. R. Kulli, Reduced elliptic and modified reduced elliptic indices of some chemical structures, Int. J. Eng. Sci. Res. Technol. 13 (2024) 11–17, https://doi.org/10.57030/ijesrt.13.2.2.2024.
[4] S. Das and V. Kumar, Investigation of closed derivation formulas for GQ and QG indices of a graph via M-polynomial, Iranian J. Math. Chem. 13 (2022) 129–144, https://doi.org/10.22052/IJMC.2022.246172.1614.
[5] S. Das and S. Rai, On closed derivation formulas of the Nirmala indices from the M-polynomial of a graph, J. Indian Chem. Soc. 100 (2023) #101017, https://doi.org/10.1016/j.jics.2023.101017.
[6] E. Deutsch and S. Klavžar, M-polynomial and degree-based topological indices, Iranian J. Math. Chem. 6 (2015) 93–102, https://doi.org/10.22052/IJMC.2015.10106.
[7] 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.
[8] S. Das and S. Rai, M-polynomial based mathematical formulation of the (a,b)-Nirmala index and its bounds, Palest. J. Math. 13 (2024) 1376–1389.
[9] A. Verma, S. Mondal, N. De and A. Pal, Topological properties of bismuth tri-iodide using neighborhood M-polynomial, Int. J. Math. Trends Technol. 65 (2019) 83–90, https://doi.org/10.14445/22315373/IJMTT-V65I10P512.
[10] H. Deng, J. Yang and F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comput. Math. Appl. 61 (2011) 3017–3023, https://doi.org/10.1016/j.camwa.2011.03.089.
[11] F. Afzal, S. Hussain, D. Afzal and S. Razaq, Some new degree based topological indices via M-polynomial, J. Inform. Optim. Sci. 41 (2020) 1061–1076, https://doi.org/10.1080/02522667.2020.1744307.
[12] B. Basavanagoud, A. P. Barangi and P. Jakkannavar, M-polynomial of some graph operations and cycle related graphs, Iranian J. Math. Chem. 10 (2019) 127–150, https://doi.org/ 10.22052/IJMC.2019.146761.1388.
[13] B. Basavanagoud and P. Jakkannavar, M-polynomial and degree-based topological indices of graphs, Electron. J. Math. Anal. Appl. 8 (2020) 75–99.
[14] 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.
[15] S. Das and V. Kumar, On M-polynomial of the two-dimensional Silicon-Carbons, Palest. J. Math. 11 (2022) 136–157.
[16] S. Das and S. Rai, M-polynomial and related degree-based topological indices of the third type of Hex-derived network, Nanosystems: Phys. Chem. Math. 11 (2020) 267–274.
[17] S. Das and S. Rai, Topological characterization of the third type of triangular Hex-derived networks, Sci. Ann. Comput. Sci. 31 (2021) 145–161, https://doi.org/10.7561/SACS.2021.2.145.
[18] S. Das and S. Rai, Degree-based topological descriptors of type 3 rectangular hex-derived networks, Bull. Inst. Combin. Appl. 95 (2022) 21–37.
[19] S. Das and S. Rai, On M-polynomial and associated topological descriptors of subdivided hex-derived network of type three, Comput. Technol. 27 (2022) 84–97.
[20] S. Hussain, A. Alsinai, D. Afzal, A. Maqbool, F. Afzal and M. Cancan, Investigation of closed formula and topological properties of remdesivir (C27H35N6O8P), Chem. Methodol. 5 (2021) 485–497.
[21] S. Das and M. Mandal, Deriving M-polynomial based topological descriptors of oral antiviral clinical drug Nirmatrelvir, Sci. Ann. Comput. Sci. 34 (2024) 113–138, https://doi.org/10.47743/SACS.2024.2.113.
[22] V. Kumar and S. Das, A novel approach to determine the Sombor-type indices via Mpolynomial, J. Appl. Math. Comput. 71 (2025) 983–1007, https://doi.org/10.1007/s12190-024-02272-4.
[23] S. Rai and S. Das, Comparative analysis of M-polynomial based topological indices between poly Hex-derived networks and their subdivisions Proyecciones 43 (2024) 69–102, https://doi.org/10.22199/issn.0717-6279-5843.
[24] S. Das, S. Rai, M. Mandal, M-polynomial and correlated topological indices of antiviral drug Molnupiravir used as a therapy for COVID-19, Polycyc. Arom. Comp. 43 (2023) 7027–7041, https://doi.org/10.1080/10406638.2022.2131854.
[25] Y. C. Kwun, A. Ali, W. Nazeer, M. A. Chaudhary and S. M. Kang, M-polynomials and degree-based topological indices of triangular, hourglass, and jagged-rectangle benzenoid systems, J. Chem. 2018 (2018) # 8213950, https://doi.org/10.1155/2018/8213950.