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Atom-bond-connectivity (ABC) indices of graphene sheets, zigzag single walled carbon nanotubes and single walled carbon nanotori

  • Soukat Ghosh , Uday Maji , Swapnadeep Mondal and Bholanath Mandal EMAIL logo
Published/Copyright: February 22, 2024
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 79 Issue 6

Abstract

Atom-bond-connectivity (ABC) indices are obtained in analytical forms for graphene sheets, zigzag single walled carbon nanotubes (SWCNTs), and single walled carbon nanotori in terms of number of rings (r) that measures the length and the number of hexagons in between two rings (h) that dictates the width of the concerned systems. The procedures followed for ABC index have been used to obtain the expressions of augmented Zagreb and Randić indices for such systems. Logarithm of ABC indices of zigzag SWCNTs are found to correlate linearly well with the bond dissociation energies per C–C bond and the Young’s moduli of said SWCNTs with fixed number of rings (r) but varying number of hexagons (h) in between two successive rings. The plot of logarithm of ABC index versus Young’s modulus of such SWCNTs in varying both r and h simultaneously is not a straight line but fits well with the sigmoidal (Boltzmann) curve. Wiener index, one of the important distance based index, has recently been found to have similar correlations with the concerned properties of such systems. Similar plots would appear for the said properties of the zigzag SWCNTs with other degree-based indices like augmented Zagreb and Randić indices, as have been indicated from their respective expressions obtained.

Keywords: degree based topological indices; atom-bond-connectivity (ABC) index; graphene; zigzag single walled carbon nanotube; single walled carbon nanotorus; bond dissociation energy per C–C bond and Young’s modulus
Corresponding author: Bholanath Mandal, Department of Chemistry, 30126 The University of Burdwan , Burdwan 713104, India, E-mail: ,

Acknowledgment

Authors thank the Reviewer for valuable suggestions.

  1. Research ethics: Maintained.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: The authors state no conflict of interest.

  4. Research funding: None declared.

  5. Data availability: Not Applicable.

References

[1] O. Ore, Graphs and Their Uses, New York, The L. W. Singer Company, 1963.Search in Google Scholar

[2] N. Deo, Graph Theory with Applications to Engineering and Computer Science, New Delhi, Prentice-Hall of India Pvt. Ltd., 1997.Search in Google Scholar

[3] N. Trinajstić, Chemical Graph Theory, Boca Raton, FL, CRC Press, 1992.Search in Google Scholar

[4] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, New York, Springer-Verlag, 1986.10.1515/9783112570180Search in Google Scholar

[5] A. T. Balaban, Ed. Topological Indices and Related Descriptors in QSAR and QSPR, The Netherlands, Gordon and Breach Science Publishers, 1999.Search in Google Scholar

[6] H. Wiener, “Structural determination of paraffin boiling points,” J. Am. Chem. Soc., vol. 69, no. 1, p. 17, 1947, https://doi.org/10.1021/ja01193a005.Search in Google Scholar PubMed

[7] H. Wiener, “Relation of the physical properties of the isomeric alkanes to molecular structure. Surface tension, specific dispersion, and critical solution temperature in aniline,” J. Phys. Chem., vol. 52, no. 6, p. 1082, 1948, https://doi.org/10.1021/j150462a018.Search in Google Scholar PubMed

[8] H. Hosoya, “Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons,” Bull. Chem. Soc. Jpn., vol. 44, p. 2332, 1971, https://doi.org/10.1246/bcsj.44.2332.Search in Google Scholar

[9] H. Hosoya, “Chemical meaning of octane number analyzed by topological indices,” Croat. Chem. Acta, vol. 75, no. 2, p. 433, 2002.Search in Google Scholar

[10] B. Mandal, M. Banerjee, and A. K. Mukherjee, “Wiener and Hosoya indices of reciprocal graphs,” Mol. Phys., vol. 103, no. 19, p. 2665, 2005, https://doi.org/10.1080/00268970500134714.Search in Google Scholar

[11] P. Ghosh, S. Basu, S. Karmakar, and B. Mandal, “Schematic generation of characteristic polynomials and the Hosoya indices of mono- and di-substituted polymer graphs of linear chains and cycles,” J. Indian Chem. Soc., vol. 91, no. 3, p. 503, 2014.Search in Google Scholar

[12] T. Ghosh, S. Mondal, and B. Mandal, “Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs,” Mol. Phys., vol. 116, no. 3, p. 361, 2018, https://doi.org/10.1080/00268976.2017.1396372.Search in Google Scholar

[13] M. Randić, “In search of structural invariants,” J. Math. Chem., vol. 9, no. 2, p. 97, 1992.10.1007/BF01164840Search in Google Scholar

[14] M. Randić, “Characterization of molecular branching,” J. Am. Chem. Soc., vol. 97, no. 23, p. 6609, 1975, https://doi.org/10.1021/ja00856a001.Search in Google Scholar

[15] A. T. Balaban, “Highly discriminating distance-based topological index,” Chem. Phys. Lett., vol. 89, no. 5, p. 399, 1982, https://doi.org/10.1016/0009-2614(82)80009-2.Search in Google Scholar

[16] I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons,” Chem. Phys. Lett., vol. 17, no. 4, p. 535, 1972, https://doi.org/10.1016/0009-2614(72)85099-1.Search in Google Scholar

[17] E. Estrada, “Atom–bond connectivity and the energetic of branched alkanes,” Chem. Phys. Lett., vol. 463, no. 4–6, p. 422, 2008, https://doi.org/10.1016/j.cplett.2008.08.074.Search in Google Scholar

[18] E. Estrada, L. Torres, L. Rodríguez, and I. Gutman, “An atom-bond connectivity index: modelling the enthalpy of formation of alkanes,” Indian J. Chem., vol. 37A, p. 849, 1998.Search in Google Scholar

[19] E. Estrada, “The ABC matrix,” J. Math. Chem., vol. 55, no. 4, p. 1021, 2017, https://doi.org/10.1007/s10910-016-0725-5.Search in Google Scholar

[20] B. Furtula, A. Graovac, and D. Vukičević, “Augmented Zagreb index,” J. Math. Chem., vol. 48, no. 2, p. 370, 2010, https://doi.org/10.1007/s10910-010-9677-3.Search in Google Scholar

[21] I. Gutman and J. Tošović, “Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices,” J. Serb. Chem. Soc., vol. 78, no. 6, p. 805, 2013, https://doi.org/10.2298/jsc121002134g.Search in Google Scholar

[22] R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, New York, Wiley, 1989.Search in Google Scholar

[23] B. Mandal, M. Banerjee, and A. K. Mukherjee, “Cardinalities of reciprocal graphs,” Int. J. Quant. Chem., vol. 99, no. 3, p. 119, 2004, https://doi.org/10.1002/qua.20120.Search in Google Scholar

[24] P. Ghosh, S. Karmakar, and B. Mandal, “Cardinalities of poly(p-phenylene) graphs,” Mol. Phys., vol. 112, no. 20, p. 2646, 2014, https://doi.org/10.1080/00268976.2014.901571.Search in Google Scholar

[25] S. C. Basak, V. R. Magnuson, G. J. Niemi, R. R. Regal, and G. D. Veith, “Topological indices: their nature, mutual relatedness, and applications,” Math. Model., vol. 8, p. 300, 1987, https://doi.org/10.1016/0270-0255(87)90594-x.Search in Google Scholar

[26] J. A. Rodríguez-Velázquez and A. T. Balaban, “Two new topological indices based on graph adjacency matrix eigenvalues and eigenvectors,” J. Math. Chem., vol. 57, no. 20, p. 1053, 2019, https://doi.org/10.1007/s10910-019-01008-1.Search in Google Scholar

[27] P. Ghosh and B. Mandal, “Characteristic polynomials of alternant edge weighted linear chains with subsequent application to some linear poly (p-phenylene) graphs,” J. Math. Chem., vol. 48, no. 4, p. 1069, 2010, https://doi.org/10.1007/s10910-010-9726-y.Search in Google Scholar

[28] P. Ghosh, S. Basu, S. Karmakar, and B. Mandal, “Matrix product forms for the characteristic polynomial coefficients of poly(p-phenylene) graphs,” J. Indian Chem. Soc., vol. 91, no. 12, p. 2197, 2014.Search in Google Scholar

[29] S. Mondal and B. Mandal, “Sum of characteristic polynomial coefficients of cycloparaphenylene graphs as topological index,” Mol. Phys., vol. 118, no. 13, p. e1685693, 2020, https://doi.org/10.1080/00268976.2019.1685693.Search in Google Scholar

[30] A. R. Katritzky, et al.., “Interpretation of quantitative structure−property and −activity relationships,” J. Chem. Inf. Comput. Sci., vol. 41, no. 3, p. 679, 2001, https://doi.org/10.1021/ci000134w.Search in Google Scholar PubMed

[31] A. Mercader, E. A. Castro, and A. A. Toropov, “QSPR modeling of the enthalpy of formation from elements by means of correlation weighting of local invariants of atomic orbital molecular graphs,” Chem. Phys. Lett., vol. 330, no. 5–6, p. 612, 2000, https://doi.org/10.1016/s0009-2614(00)01126-x.Search in Google Scholar

[32] M. Randić, M. Nović, and D. Plavšić, Solved and Unsolved Problems of Structural Chemistry, Boca Raton, CRC Press, 2016.Search in Google Scholar

[33] M. Barysz, D. Plavšić, and N. Trinajstić, “A note on topological indices,” MATCH Commun. Math. Comput. Chem., vol. 19, p. 89, 1986.Search in Google Scholar

[34] M. A. Johnson and G. M. Maggiora, Concepts and Applications of Molecular Similarity, New York, Wiley Interscience, 1990.Search in Google Scholar

[35] S. Hayat, N. Suhaili, and H. Jamil, “Statistical significance of valency-based topological descriptors for correlating thermodynamic properties of benzenoid hydrocarbons with applications,” Comput. Theor. Chem., vol. 1227, p. 114259, 2023, https://doi.org/10.1016/j.comptc.2023.114259.Search in Google Scholar

[36] S. Hayat, S. Khan, A. Khan, and J.-B. Liu, “Valency-based molecular descriptors for measuring the π-electronic energy of lower polycyclic aromatic hydrocarbons,” Polycyl. Aromat. Comp., vol. 42, no. 4, p. 1113, 2022, https://doi.org/10.1080/10406638.2020.1768414.Search in Google Scholar

[37] M. Y. H. Malik, M. A. Binyamin, and S. Hayat, “Correlation ability of degree-based topological indices for physicochemical properties of polycyclic aromatic hydrocarbons with applications,” Polycyl. Aromat. Comp., vol. 42, no. 9, p. 6267, 2022, https://doi.org/10.1080/10406638.2021.1977349.Search in Google Scholar

[38] D. J. Wild and P. Willett, “Similarity searching in files of three-dimensional chemical structures. Implementation of atom mapping on the distributed array processor DAP-610, the MasPar MP-1104, and the connection machine CM-200,” J. Chem. Inf. Comput. Sci., vol. 34, no. 1, p. 224, 1994, https://doi.org/10.1021/ci00017a029.Search in Google Scholar

[39] S. C. Basak, B. D. Gute, and A. T. Balaban, “Interrelationship of Major Topological Indices Evidenced by Clustering,” Croat. Chem. Acta, vol. 77, no. 1–2, p. 331, 2004.Search in Google Scholar

[40] S. Hayat, “Computing distance-based topological descriptors of complex chemical networks: new theoretical techniques,” Chem. Phys. Lett., vol. 688, p. 51, 2017, https://doi.org/10.1016/j.cplett.2017.09.055.Search in Google Scholar

[41] S. Hayat, M. Imran, and J.-B. Liu, “An efficient computational technique for degree and distance based topological descriptors with applications,” IEEE Access, vol. 7, p. 32276, 2019, https://doi.org/10.1109/access.2019.2900500.Search in Google Scholar

[42] S. Hayat, S. Khan, and M. Imran, “Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones,” Arab. J. Chem., vol. 14, no. 3, p. 102994, 2021, https://doi.org/10.1016/j.arabjc.2021.102994.Search in Google Scholar

[43] S. Hayat, S. Khan, A. Khan, and M. Imran, “A computer-based method to determine predictive potential of distance-spectral descriptors for measuring the π-electronic energy of benzenoid hydrocarbons with applications,” IEEE Access, vol. 9, p. 19238, 2021, https://doi.org/10.1109/access.2021.3053270.Search in Google Scholar

[44] D. J. Klein and B. Mandal, ““Pure-polyhex” π-networks: topo-combinatorics,” Croat. Chem. Acta, vol. 93, no. 4, p. 349, 2020, https://doi.org/10.5562/cca3790.Search in Google Scholar

[45] D. J. Klein, “Graphitic polymer strips with edge states,” Chem. Phys. Lett., vol. 217, no. 3, p. 261, 1994, https://doi.org/10.1016/0009-2614(93)e1378-t.Search in Google Scholar

[46] K. S. Novoselov, et al.., “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. U.S.A., vol. 102, no. 30, p. 10451, 2005, https://doi.org/10.1073/pnas.0502848102.Search in Google Scholar PubMed PubMed Central

[47] K. S. Novoselov, et al.., “Electric field effect in atomically thin carbon films,” Science, vol. 306, no. 5696, p. 666, 2004, https://doi.org/10.1126/science.1102896.Search in Google Scholar PubMed

[48] K. S. Novoselov, et al.., “Two-dimensional gas of massless Dirac fermions in graphene,” Nature, vol. 438, no. 7065, p. 197, 2005, https://doi.org/10.1038/nature04233.Search in Google Scholar PubMed

[49] A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater., vol. 6, no. 3, p. 183, 2007, https://doi.org/10.1038/nmat1849.Search in Google Scholar PubMed

[50] K. Y. Rhee, Electronic and Thermal Properties of Graphene, Basel, Switzerland, MDPI, 2019.Search in Google Scholar

[51] C. N. Rao, A. K. Sood, K. S. Subrahmanyam, and A. Govindaraj, “Graphene: the new two-dimensional nanomaterial,” Angew. Chem., Int. Ed., vol. 48, no. 42, p. 7752, 2009, https://doi.org/10.1002/anie.200901678.Search in Google Scholar PubMed

[52] K. Cao, S. Feng, Y. Han, L. Gao, T. H. Ly, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun., vol. 11, p. 284, 2020.10.1038/s41467-019-14130-0Search in Google Scholar PubMed PubMed Central

[53] W. Cai, et al.., “Thermal transport in suspended and supported monolayer graphene grown by chemical vapor deposition,” Nano Lett., vol. 10, no. 5, p. 1645, 2010, https://doi.org/10.1021/nl9041966.Search in Google Scholar PubMed

[54] G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Terrones, “Electronic and optical properties of strained graphene and other strained 2D materials: a review,” Rep. Prog. Phys., vol. 80, no. 9, p. 096501, 2017, https://doi.org/10.1088/1361-6633/aa74ef.Search in Google Scholar PubMed

[55] E. W. Hill, A. Vijayaragahvan, and K. Novoselov, “Graphene Sensors,” IEEE Sens. J., vol. 11, no. 12, p. 3161, 2011.10.1109/JSEN.2011.2167608Search in Google Scholar

[56] Q. Han, et al.., “Graphene biodevices for early disease diagnosis based on biomarker detection,” ACS Sens., vol. 6, no. 11, p. 3841, 2021, https://doi.org/10.1021/acssensors.1c01172.Search in Google Scholar PubMed

[57] P. Ghosh, D. J. Klein, and B. Mandal, “Analytical eigenspectra of alternant edge-weighted graphs of linear chains and cycles: some applications,” Mol. Phys., vol. 112, no. 16, p. 2093, 2014, https://doi.org/10.1080/00268976.2014.886737.Search in Google Scholar

[58] S. Iijima, “Helical microtubules of graphitic carbon,” Nature, vol. 354, no. 6348, p. 56, 1991, https://doi.org/10.1038/354056a0.Search in Google Scholar

[59] S. Iijima and I. Toshinari, “Single-shell carbon nanotubes of 1-nm diameter,” Nature, vol. 363, no. 6430, p. 603, 1993, https://doi.org/10.1038/363603a0.Search in Google Scholar

[60] M. S. Dresselhaus, G. Dresselhaus, and R. Saito, “Physics of carbon nanotubes,” Carbon, vol. 33, no. 7, p. 883, 1995, https://doi.org/10.1016/0008-6223(95)00017-8.Search in Google Scholar

[61] M. F. L. De Volder, S. H. Tawfick, R. H. Baughman, and A. J. Hart, “Carbon nanotubes: present and future commercial applications,” Science, vol. 339, no. 6119, p. 535, 2013, https://doi.org/10.1126/science.1222453.Search in Google Scholar PubMed

[62] M. Zhang, et al.., “Strong, transparent, multifunctional, carbon nanotube sheets,” Science, vol. 309, no. 5738, p. 1215, 2005, https://doi.org/10.1126/science.1115311.Search in Google Scholar PubMed

[63] A. B. Dalton, et al.., “Super-tough carbon-nanotube fibres,” Nature, vol. 423, no. 6941, p. 703, 2003, https://doi.org/10.1038/423703a.Search in Google Scholar PubMed

[64] K. Koziol, et al.., “High-performance carbon nanotube fiber,” Science, vol. 318, no. 5858, p. 1892, 2007, https://doi.org/10.1126/science.1147635.Search in Google Scholar PubMed

[65] C. D. Zeinalipour-Yazdi, E. Z. Loizidou, and A. Chutia, “Size-dependent bond dissociation enthalpies in single-walled carbon nanotubes,” Chem. Phys. Lett., vol. 731, p. 136628, 2019, https://doi.org/10.1016/j.cplett.2019.136628.Search in Google Scholar

[66] Q. W. Ahmed, D. A. Alazawi, and H. B. Mohammed, “Study the behavior of elastic modulus for zigzag and armchair single wall carbon nanotube structure with FEM,” J. Eng. Sustain. Dev., vol. 25, no. 4, p. 114, 2021.10.31272/jeasd.25.4.10Search in Google Scholar

[67] B. WenXing, Z. ChangChun, and C. WanZhao, “Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics,” Phys. B Cond. Mat., vol. 352, no. 1–4, p. 156, 2004, https://doi.org/10.1016/j.physb.2004.07.005.Search in Google Scholar

[68] J. R. Xiao, B. A. Gama, and G. W. GillespieJr., “An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes,” Int. J. Solids Struct., vol. 42, no. 11–12, p. 3075, 2005, https://doi.org/10.1016/j.ijsolstr.2004.10.031.Search in Google Scholar

[69] T. Ghosh and B. Mandal, “Wiener indices of zigzag single walled carbon nanotubes and related nanotories,” Chem. Phys., vol. 572, p. 111973, 2023, https://doi.org/10.1016/j.chemphys.2023.111973.Search in Google Scholar

[70] M. Ahlskog, et al.., “Ring formations from catalytically synthesized carbon nanotubes,” Chem. Phys. Lett., vol. 300, no. 1–2, p. 202, 1999, https://doi.org/10.1016/s0009-2614(98)01322-0.Search in Google Scholar

[71] M. Sano, A. Kamino, J. Okamura, and S. Shinkai, “Ring closure of carbon nanotubes,” Science, vol. 293, no. 5333, p. 1299, 2001, https://doi.org/10.1126/science.1061050.Search in Google Scholar PubMed

[72] L. Liu, G. Y. Guo, C. S. Jayanthi, and S. Y. Wu, “Colossal paramagnetic moments in metallic carbon nanotori,” Phys. Rev. Lett., vol. 88, no. 21, p. 217206, 2002, https://doi.org/10.1103/physrevlett.88.217206.Search in Google Scholar
Received: 2023-10-09
Accepted: 2024-01-29
Published Online: 2024-02-22
Published in Print: 2024-06-25

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Atomic, Molecular & Chemical Physics
  3. Atom-bond-connectivity (ABC) indices of graphene sheets, zigzag single walled carbon nanotubes and single walled carbon nanotori
  4. Calibration-free approaches for quantitative analysis of a brass sample
  5. Dynamical Systems & Nonlinear Phenomena
  6. Significance of hafnium nanoparticles in hydromagnetic non-Newtonian fluid-particle suspension model through divergent channel with uniform heat source: thermal analysis
  7. Evolution of shock waves in dusty nonideal gas flow with magnetic field
  8. Quantum Theory
  9. Analysis of microstrip low pass filter at terahertz frequency range in finite difference time domain method for radar applications
  10. Solid State Physics & Materials Science
  11. Enhancing charge transport and photoluminescence characteristics via transition metals doping in ITO thin films
  12. Effect of zinc doping on structural, bonding nature and magnetic properties of co-precipitated magnesium–nickel ferrites
  13. Nonreciprocal transmission in composite structure with Weyl semimetal defect layer
https://doi.org/10.1515/zna-2023-0277
Keywords for this article
degree based topological indices; atom-bond-connectivity (ABC) index; graphene; zigzag single walled carbon nanotube; single walled carbon nanotorus; bond dissociation energy per C–C bond and Young’s modulus

Articles in the same Issue

  1. Frontmatter
  2. Atomic, Molecular & Chemical Physics
  3. Atom-bond-connectivity (ABC) indices of graphene sheets, zigzag single walled carbon nanotubes and single walled carbon nanotori
  4. Calibration-free approaches for quantitative analysis of a brass sample
  5. Dynamical Systems & Nonlinear Phenomena
  6. Significance of hafnium nanoparticles in hydromagnetic non-Newtonian fluid-particle suspension model through divergent channel with uniform heat source: thermal analysis
  7. Evolution of shock waves in dusty nonideal gas flow with magnetic field
  8. Quantum Theory
  9. Analysis of microstrip low pass filter at terahertz frequency range in finite difference time domain method for radar applications
  10. Solid State Physics & Materials Science
  11. Enhancing charge transport and photoluminescence characteristics via transition metals doping in ITO thin films
  12. Effect of zinc doping on structural, bonding nature and magnetic properties of co-precipitated magnesium–nickel ferrites
  13. Nonreciprocal transmission in composite structure with Weyl semimetal defect layer
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