Topological Indices of Advanced Nanotube Architectures: Hexagonal, Zigzag-Edge Coronoid and Starphene Fusions
Abstract
Chemical graph theory (CGT) is a pivotal field of modern science where graph theory is employed to address complex chemical challenges. By leveraging topological indices (TIs), it is possible to predict numerous properties of chemical structures without the need for experimental procedures. However, the computation of TIs often demands significant effort. In this study, we introduce a streamlined approach to compute TIs efficiently. Initially, we derive NM-polynomials for hexagonal parallelogram nanotubes, triangular benzenoids, and zigzag-edge coronoids fused with starphene nanotubes. Subsequently, we calculate various neighborhood-based TIs for these structures using critical properties derived from the NM-polynomial. The results of this study hold significant potential for applications in drug design, pharmaceuticals, and applied physics.
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References
[1] Kang SM, Zahid MA, Nazeer W, Gao W. Calculating the degree-based topological indices of dendrimers. Open Chemistry. 2018 Jan 1;16(1):681-8.
[2] Gao W, Younas M, Farooq A, Virk AU, Nazeer W. Some reverse degree-based topological indices and polynomials of dendrimers. Mathematics. 2018 Oct 22;6(10):214.
[3] Yan F, Shang Q, Xia S, Wang Q, Ma P. Application of topological index in predicting ionic liquids densities by the quantitative structure property relationship method. Journal of Chemical & Engineering Data. 2015 Mar 12;60(3):734-9.
[4] Hanson MP, Rouvray DH. Novel applications of topological indices. 2. Prediction of the threshold soot index for hydrocarbon fuels. Journal of Physical Chemistry. 1987May;91(11):2981-5.
[5] Mukhtar Ahmad, Ather Qayyum, Gulnaz Atta, Siti Suzlin Supadi, Muhammad Saleem, Usman Ali, A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application, International journal of analysis and applications, 22; 66 (2024).
[6] Mukhtar Ahmad, Ather Qayyum, Gulnaz Atta, Siti Suzlin Supadi, Muhammad Saleem, Usman Ali, A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application, International journal of analysis and applications, 22; 66 (2024).
[7] 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 Indices, International journal of analysis and applications, 21;66 (2023).
[8] R. M. K. Iqbal, M. Ahmad, A. Qayyum*, S. S. Supadi, M. J. Hussain, S. Raza, On Degree Based Topological Indices of Toeplitz Graphs, International journal of analysis and applications, 21;111 (2023).
[9] Kiralj R, Ferreira M. Basic validation procedures for regression models in QSAR and QSPR studies: theory and application. Journal of the Brazilian Chemical Society. 2009;20:770-87.
[10] T Stanton D. QSAR and QSPR model interpretation using partial least squares (PLS) analysis. Current computer-aided drug design. 2012 Jun 1;8(2):107-27.
[11] Wiener H. Structural determination of paraffin boiling points. Journal of the American chemical society. 1947 Jan;69(1):17-20.
[12] Randic M. Characterization of molecular branching. Journal of the American Chemical Society. 1975 Nov;97(23):6609-15.
[13] Gutman I, Furtula B, Katanic V. Randic index and information. AKCE International Journal of Graphs and Combinatorics. 2018 Dec 1;15(3):307-12.
[14] Randi´c M. On history of the Randic index and emerging hostility toward chemical graph theory. MATCH Commun. Math. Comput. Chem. 2008;59(5).
[15] Gutman I. An exceptional property of first Zagreb index. MATCH Commun. Math. Comput. Chem. 2014 Jan 1;72(3):733-40.
[16] Zhou B, Gutman I. Further properties of Zagreb indices. MATCH Commun. Math. Comput. Chem. 2005 Nov 9;54(1):233-9.
[17] Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem. 2004 Feb 1;50(1):83-92.
[18] Ghorbani M, Jalali M. Computing a new topological index of nano structures. Digest Journal of Nanomaterials and Biostructures. 2009 Dec 1;4(4):681-5.
[19] Brendl´e E, Papirer E. A new topological index for molecular probes used in inverse gas chromatography for the surface nanorugosity evaluation. Journal of colloid and interface science. 1997 Oct 1;194(1):207-16.
[20] Deutsch E, Klav¸zar S. M-polynomial and degree-based topological indices. arXiv preprint arXiv:1407.1592. 2014 Jul 7.
[21] Munir M, Nazeer W, Rafique S, Kang SM. M-polynomial and related topological indices of nanostar dendrimers. Symmetry. 2016 Sep 21;8(9):97.
[22] Kwun YC, Ali A, Nazeer W, Ahmad Chaudhary M, Kang SM. M-polynomials and degree based topological indices of triangular, hourglass, and jagged-rectangle benzenoid systems. Journal of Chemistry. 2018 Jan 1;2018.
[23] Mondal S, Siddiqui MK, De N, Pal A. Neighborhood M-polynomial of crystallographic structures. Biointerface Res. Appl. Chem. 2021;11(2):9372-81.
[24] Ghorbani M, Hosseinzadeh MA. The third version of Zagreb index. Discrete mathematics, algorithms and applications. 2013 Dec 13;5(04):1350039.
[25] 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.
[26] 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.
[27] 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
[28] 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.
[29] Wu J, Nazeer S, Ahmed I, Yasmin F. Study of Graphene Networks and Line Graph of Graphene Networks via NM-polynomial and Topological Indices. Journal of Mathematics. 2022 Nov 12;2022.
[2] Gao W, Younas M, Farooq A, Virk AU, Nazeer W. Some reverse degree-based topological indices and polynomials of dendrimers. Mathematics. 2018 Oct 22;6(10):214.
[3] Yan F, Shang Q, Xia S, Wang Q, Ma P. Application of topological index in predicting ionic liquids densities by the quantitative structure property relationship method. Journal of Chemical & Engineering Data. 2015 Mar 12;60(3):734-9.
[4] Hanson MP, Rouvray DH. Novel applications of topological indices. 2. Prediction of the threshold soot index for hydrocarbon fuels. Journal of Physical Chemistry. 1987May;91(11):2981-5.
[5] Mukhtar Ahmad, Ather Qayyum, Gulnaz Atta, Siti Suzlin Supadi, Muhammad Saleem, Usman Ali, A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application, International journal of analysis and applications, 22; 66 (2024).
[6] Mukhtar Ahmad, Ather Qayyum, Gulnaz Atta, Siti Suzlin Supadi, Muhammad Saleem, Usman Ali, A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application, International journal of analysis and applications, 22; 66 (2024).
[7] 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 Indices, International journal of analysis and applications, 21;66 (2023).
[8] R. M. K. Iqbal, M. Ahmad, A. Qayyum*, S. S. Supadi, M. J. Hussain, S. Raza, On Degree Based Topological Indices of Toeplitz Graphs, International journal of analysis and applications, 21;111 (2023).
[9] Kiralj R, Ferreira M. Basic validation procedures for regression models in QSAR and QSPR studies: theory and application. Journal of the Brazilian Chemical Society. 2009;20:770-87.
[10] T Stanton D. QSAR and QSPR model interpretation using partial least squares (PLS) analysis. Current computer-aided drug design. 2012 Jun 1;8(2):107-27.
[11] Wiener H. Structural determination of paraffin boiling points. Journal of the American chemical society. 1947 Jan;69(1):17-20.
[12] Randic M. Characterization of molecular branching. Journal of the American Chemical Society. 1975 Nov;97(23):6609-15.
[13] Gutman I, Furtula B, Katanic V. Randic index and information. AKCE International Journal of Graphs and Combinatorics. 2018 Dec 1;15(3):307-12.
[14] Randi´c M. On history of the Randic index and emerging hostility toward chemical graph theory. MATCH Commun. Math. Comput. Chem. 2008;59(5).
[15] Gutman I. An exceptional property of first Zagreb index. MATCH Commun. Math. Comput. Chem. 2014 Jan 1;72(3):733-40.
[16] Zhou B, Gutman I. Further properties of Zagreb indices. MATCH Commun. Math. Comput. Chem. 2005 Nov 9;54(1):233-9.
[17] Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem. 2004 Feb 1;50(1):83-92.
[18] Ghorbani M, Jalali M. Computing a new topological index of nano structures. Digest Journal of Nanomaterials and Biostructures. 2009 Dec 1;4(4):681-5.
[19] Brendl´e E, Papirer E. A new topological index for molecular probes used in inverse gas chromatography for the surface nanorugosity evaluation. Journal of colloid and interface science. 1997 Oct 1;194(1):207-16.
[20] Deutsch E, Klav¸zar S. M-polynomial and degree-based topological indices. arXiv preprint arXiv:1407.1592. 2014 Jul 7.
[21] Munir M, Nazeer W, Rafique S, Kang SM. M-polynomial and related topological indices of nanostar dendrimers. Symmetry. 2016 Sep 21;8(9):97.
[22] Kwun YC, Ali A, Nazeer W, Ahmad Chaudhary M, Kang SM. M-polynomials and degree based topological indices of triangular, hourglass, and jagged-rectangle benzenoid systems. Journal of Chemistry. 2018 Jan 1;2018.
[23] Mondal S, Siddiqui MK, De N, Pal A. Neighborhood M-polynomial of crystallographic structures. Biointerface Res. Appl. Chem. 2021;11(2):9372-81.
[24] Ghorbani M, Hosseinzadeh MA. The third version of Zagreb index. Discrete mathematics, algorithms and applications. 2013 Dec 13;5(04):1350039.
[25] 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.
[26] 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.
[27] 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
[28] 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.
[29] Wu J, Nazeer S, Ahmed I, Yasmin F. Study of Graphene Networks and Line Graph of Graphene Networks via NM-polynomial and Topological Indices. Journal of Mathematics. 2022 Nov 12;2022.
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2025-08-24
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