Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

M-polynomials and topological indices of linear chains of benzene, napthalene and anthracene

1 Techanical and Electrical Engineering Department, Anqing Vocational and Technical College, Anqing 246003, P.R. China
2 Teaching Department of Public Basic Course, Anhui International University, Hefei 231201, China
3 Department of Mathematics, Division of Science and Technology, University of Education, Lahore-54590, Pakistan
4 School of Mathematics, Southeast University, Nanjing 210096, China

Special Issues: Theoretical researches in materials and condensed matter physics

The universality of M-polynomial paves way towards establishing closed forms of many leading degree-based topological indices as it is done by Hosoya polynomial for distance-based indices. The study of topological indices is recently one of the most active research areas in chemical graph theory. The aim of this paper is to establish closed formulas for M-polynomials of Linear chains of benzene, napthalene, and anthracene graphs. From this polynomial we also compute as many as nine degree-based topological indices for these three chains. Our results will potentially play an important role in pharmacy, drug design, and many other applied areas of molecular sciences.
  Article Metrics

Keywords M-polynomial; degree-based index; linear chain of benzene; napthalene; anthracene

Citation: Cheng-Peng Li, Cheng Zhonglin, Mobeen Munir, Kalsoom Yasmin, Jia-bao Liu. M-polynomials and topological indices of linear chains of benzene, napthalene and anthracene. Mathematical Biosciences and Engineering, 2020, 17(3): 2384-2398. doi: 10.3934/mbe.2020127


  • 1. G. Rucker, C. Rucker, On topological indices, boiling points, and cycloalkanes, J. Chem. Inf. Comput. Sci., 39 (1999), 788-802.
  • 2. S. Klavzar, I. Gutman, A comparison of the schultz molecular topological index with the wiener index, J. Chem. Inf. Comput. Sci., 36 (1996), 1001-1003.
  • 3. F. M. Bruckler, C. T. Dosli, A. Graovac, I. Gutman, On a class of distance-based molecular structure descriptors, Chem. Phys. Lett., 503 (2011), 336-338.
  • 4. H. Deng, J. Yang, F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comput. Math. Appl., 61 (2011), 3017-3023.
  • 5. H. Zhang, F. Zhang, The clar covering polynomial of hexagonal systems I, Discrete Appl. Math., 69 (1996), 147-167.
  • 6. I. Gutman, Some properties of the Wiener polynomial, Graph. Theory Notes N. Y., 125 (1993), 13-18.
  • 7. E. Deutsch, S. Klavzar, M-polynomial and degreebased topological indices,Iran. J. Math. Chem, 6 (2015), 93-102.
  • 8. M. Munir, W. Nazeer, S. Rafique, S. M. Kang, M-polynomial and related topological indices of nanostar dendrimers, Symmetry, 8 (2016), 97.
  • 9. M. Munir, W.Nazeer, S.Rafique, A. R. Nizami, S. M. Kang, M-polynomial and degree-based topological indices of titaniananotubes, Symmetry, 8 (2016), 117.
  • 10. Y. Kwun, M. Munir, W. Nazeer, S. Rafique, S. M. Kang, M-polynomial and degree-based topological indices of Vphenylenic nanotubes and nanotori, Sci. Rep., 7 (2017), 8756.
  • 11. M. Munir, W. Nazeer, S. Rafique, A. R. Nizami, S. M. Kang, Some computational aspects of triangular boron nanotubes, Symmetry, 9 (2016), 6.
  • 12. M. Munir, W. Nazeer, S. Rafique, S. M. Kang, M-polynomial and degree-based topological indices of polyhex nanotubes, Symmetry, 8 (2016), 149.
  • 13. A. Ali, W. Nazeer, M. Munir, S. M. Kang, M-Polynomials and topological indices of zigzag and rhombic benzenoidsystems, Open Chem., 16 (2018), 73-78.
  • 14. H. Wiener, Structural determination of paran boiling points, JACS, 69 (1947), 17-20.
  • 15. A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math., 66 (2001), 211-249.
  • 16. I. Gutman, O. E. Polansky, Mathematical concepts in organic chemistry, Springer Science and Business Media, Berlin, Germany, (2012).
  • 17. M. Randic, On characterization of molecular attributes, Acta Chim. Slov., 45 (1998), 239.
  • 18. B. Bollobas, P. Erdos, Graphs of extremal weights, Ars Combinatoria, 50 (1998), 225-233.
  • 19. D. Amic, D. Beslo, B. Lucic, S. Nikolic, N. Trinajstic, The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci., 38 (1998), 819-822.
  • 20. Y. Hu, X. Li, Y. Shi, T. Xu, I.Gutman, On molecular graphs with smallest and greatest zeroth-order general Randic index, Match-Commun. Math. Co., 54 (2005), 425-434.
  • 21. G. Caporossi, I. Gutman, P. Hansen, L. Pavlovic, Graphs with maximum connectivity index, Comput. Biol. Chem., 27(2003), 85-90.
  • 22. X. Li, I. Gutman, Mathematical chemistry monographs no. 1, Kragujevac, 330 (2006), 6.
  • 23. L. Kier, Molecular connectivity in chemistry and drug research, Elsevier, NY. USA, 14 (2012).
  • 24. L. B. Kierand, L. H. Hall, Molecular connectivity in structure activity analysis, research studies, chemometrics research studies press, 1st edition, (1986).
  • 25. I. Gutman, B. Furtula, Recent Results in the Theory of Randic Index, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, Serbia, (2008).
  • 26. S. Nikolic, G. Kovacevic, A. Milicevic, N. Trinajstic, The Zagreb indices 30 years after, Croa. Chem. Acta, 76 (2003), 113-124.
  • 27. I. Gutman, K. C. Das, The first Zagreb index 30 years after, Math. Commun. Math. Comput. Chem., 50 (2004), 83-92.
  • 28. K. C. Das, I. Gutman, Some properties of the second Zagreb index, Match-Commun. Math. Co., 52 (2004), 1-3.
  • 29. N. Trinajstic, S. Nikolic, A. Milicevic, I. Gutman, About the Zagreb Indices, Kemija u Industriji: Casopis Kemicara i Kemijskih Inzenjera Hrvatske, 59 (2010), 577-589.
  • 30. D. Vukicevic, A. Graovac, Valence connectivity versus Randic, Zagreb and modified Zagreb index, a linear algorithm to check discriminative properties of indices in acyclic molecular graphs, Croa. Chem. Acta, 77 (2004), 501-508.
  • 31. T. Doslic, I. Zubac, Saturation number of benzenoid graphs, Math. Commun. Math. Comput. Chem., (2015).
  • 32. A. Graovac, N. Trinjstic, Graph thoretical search for benzenoid polymers with zero energy gap, Croat. Chem. Acta, 4 (1981), 571-579.
  • 33. I. Gutman, M. Petkovsek, R. Z. pleterselt, On Hosoya polynomial of benzenoid Graph, Math. Commun., (2001).
  • 34. D. Vukicevic, N. Trinajstic, Wiener indices of benzenoid graph, Bull. Chem. Technol. Macedon., 23 (2004), 113-129.
  • 35. I. Gutman, S. Radenkovic, Simple formula for calculating resonance energy of benzenoid hydrocarbons, Bull. Chem. Technol. Macedon., 25 (2006), 17-21.
  • 36. A. vesel, 4-tilings of benzenoid graphs Math. Commun Math. Comput. Chem., 62 (2009), 221-234.
  • 37. K. C. Das, F. M. Bhatti, S. G. lee, I. Gutman, Spectral properties of the he matrix of hexagonal systems, Math. Commun Math. Comput. Chem., 65 (2011), 753-774.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved