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Distance based indices in nanotubical graphs: part 3

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Abstract

Nanotubical graphs are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. In this paper we derive asymptotics for generalized Wiener index for such graphs. More generally, we show that if \(I^{\lambda }(G)=\sum _{u\ne v}f(u,v)\mathrm{dist}^{\lambda }(u,v)\), where \(\lambda \ge -1\) and f(u, v) is a nonnegative symmetric function which is increasing and which depends only on \(\deg (u)\) and \(\deg (v)\), then the leading term depend only on \(\lambda \), f(x, y) when \(\deg (x)=\deg (y)=3\), and the circumference of the nanotube. This general form determines the asymptotics also for some other indices as Hararay index, additively weighted Harary index, generalized degree distance, modified generalized degree distance, and possibly others of this kind.

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References

  1. Y. Alizadeh, A. Iranmanesh, T. Doslic, Additively weighted Harary index of some composite graphs. Discrete Math. 313, 26–34 (2013)

    Article  Google Scholar 

  2. V. Andova, D. Blenkuš, T. Došlić, F. Kardoš, R. Škrekovski, On diameter of nanotubical fullerene graphs. MATCH Commun. Math. Comput. Chem. 73, 529–542 (2015)

    Google Scholar 

  3. V. Andova, F. Kardoš, R. Škrekovski, Mathematical aspects of Fullerenes. ARS Math. Contemp. 11, 353–379 (2016)

    Article  Google Scholar 

  4. V. Andova, M. Knor, R. Škrekovski, Distances on nanotubical structures. J. Math. Chem. 54, 1575–1584 (2016)

    Article  CAS  Google Scholar 

  5. V. Andova, M. Knor, R. Škrekovski, Distances based topological indices: part 1. J. Math. Chem. 56, 2801–2815 (2018)

    Article  CAS  Google Scholar 

  6. V. Andova, M. Knor, R. Škrekovski, Distances based topological indices: part 2. J. Math. Chem. 56, 3076–3088 (2018)

    Article  CAS  Google Scholar 

  7. A.T. Balaban, Topological indices based on topological distances in molecular graphs. Pure Appl. Chem. 55, 199–206 (1983)

    Article  CAS  Google Scholar 

  8. A.T. Balaban, A comparison between various topological indices, particularly between the index J and Wiener index W, in Topology in Chemistry Discrete Mathematics of Molecules, ed. by D.H. Rouvray, R.B. King (Horwood, Chichester, 2002), pp. 89–112

    Google Scholar 

  9. A.T. Balaban, P.V. Khadikar, C.T. Supuran, A. Thakur, M. Thakur, Study on supramolecular complexing ability vis’a-vis estimation of \(pK(a)\) of substituted sulfonamides: dominating role of Balaban index (\(J\)). Bioorg. Med. Chem. Lett. 15, 3966–3973 (2005)

    Article  CAS  Google Scholar 

  10. H. Deng, On the Sum-Balaban Index. MATCH Commun. Math. Comput. Chem. 66, 273–284 (2011)

    CAS  Google Scholar 

  11. J. Devillers, A.T. Balaban (eds.), Topological Indices and Related Descriptors in QSAR and QSPR (Gordon & Breach, Amsterdam, 1999)

    Google Scholar 

  12. G. Grassy, B. Calas, A. Yasri, R. Lahana, J. Woo, S. Iyer, M. Kaczorek, F. Floćh, R. Buelow, Computer-assisted rational design of immunosuppressive compounds. Nat. Biotechnol. 16, 748–752 (1998)

    Article  CAS  Google Scholar 

  13. I. Gutman, A property of the Wiener number and its modifications. Indian J. Chem. 36A, 128–132 (1997)

    CAS  Google Scholar 

  14. I. Gutman, A.A. Dobrynin, S. Klavzar, L. Pavlovic, Wiener-type invariants of trees and their relation. Bull. Inst. Combin. Appl. 40, 23–30 (2004)

    Google Scholar 

  15. O. Ivanciuc, T.S. Balaban, A.T. Balaban, Design of topological indices, part 4, reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem. 12, 309–318 (1993)

    Article  CAS  Google Scholar 

  16. A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh, M.V. Diudea, Generalized degree distance of trees, unicyclic and bicyclic graphs. Studia Ubb Chemia LVII 4, 73–85 (2012)

    Google Scholar 

  17. P.V. Khadikar, C.T. Supuran, A. Thakur, M. Thakur, QSAR study on benzene- sulphonamide carbonic anhydrase inhibitors: topological approach using Balaban index. Bioorg. Med. Chem. 12, 789–793 (2004)

    Article  Google Scholar 

  18. M. Knor, A. Tepeh, R. Škrekovski, Balaban index of cubic graphs. MATCH Commun. Math. Comput. Chem. 73, 519–528 (2015)

    Google Scholar 

  19. H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, C60: buckminsterfullerene. Nature 318, 162–163 (1985)

    Article  CAS  Google Scholar 

  20. E.A. Laird, F. Kuemmeth, G. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, L.P. Kouwenhoven, Quantum transport in carbon nanotubes. Rev. Mod. Phys. 87, 703–771 (2015)

    Article  CAS  Google Scholar 

  21. X. Lu, Z. Chen, Curved pi-conjugation, aromaticity, and the related chemistry of small fullerenes (C60) and single-walled carbon nanotubes. Chem. Rev. 105, 3643–3696 (2005)

    Article  CAS  Google Scholar 

  22. D. Plavšić, S. Nikolić, N. Trinajstić, Z. Mihalić, On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12, 235–250 (1993)

    Article  Google Scholar 

  23. H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 1, 17–20 (1947)

    Article  Google Scholar 

  24. H.Y. Zhu, D.J. Klenin, I. Lukovits, Extensions of the Wiener number. J. Chem. Inf. Comput. Sci. 36, 420–428 (1996)

    Article  CAS  Google Scholar 

  25. X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R.O. Jones, Y. Ando, Smallest carbon nanotube is 3 Å in diameter. Phys. Rev. Lett. 92, 125502-1 (2004)

    Google Scholar 

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Acknowledgements

The authors acknowledge partial support by Slovak research Grants VEGA 1/0142/17, VEGA 1/0238/19, APVV-15-0220, APVV-17-0428, Slovenian research agency ARRS, Program No. P1-0383, and National Scholarship Programme of the Slovak Republic.

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Authors and Affiliations

  1. Faculty of Electrical Engineering and Information Technologies, Ss Cyril and Methodius University, Ruger Boskovik 18, 1000, Skopje, Macedonia

    Vesna Andova

  2. Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovakia

    Martin Knor

  3. Faculty of Information Studies, Novo mesto & FMF, University of Ljubljana, Ljubljana, Slovenia

    Riste Škrekovski

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  1. Vesna Andova
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  2. Martin Knor
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Correspondence to Vesna Andova.

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Andova, V., Knor, M. & Škrekovski, R. Distance based indices in nanotubical graphs: part 3. J Math Chem 59, 250–263 (2021). https://doi.org/10.1007/s10910-020-01192-5

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  • DOI: https://doi.org/10.1007/s10910-020-01192-5

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