Bounds of the Symmetric Division Deg Index for Trees and Unicyclic Graphs with a Perfect Matching

Document Type : Research Paper

Authors

Department of Mathematical Sciences, IIT (BHU) Varanasi

Abstract

The Symmetric division deg (SDD) index is a well-established valuable index in the analysis of quantitative structure-property and structure-activity relationships for molecular graphs. In this paper, we study the range of SDD-index for special classes of trees and unicyclic graphs. We present the first four lower bounds for SDD-index of trees and unicyclic graphs, which admit a perfect matching and find the subclasses of graphs that attain these bounds. Further, we also compute the upper bounds of SDD-index for the collection of molecular graphs, namely the trees and unicyclic graphs, each having maximum degree four and that admit a perfect matching.

Keywords

References

  1. A. Ali, S. Elumalai and T. Mansour, On the symmetric division deg index of molecular graphs, MATCH Commun. Math. Comput. Chem. 83 (2020) 205–220.
  2. S. C. Basak, Use of graph invariants in quantitative structure-activity relationship studies, Croat. Chem. Acta 89 (4) (2016) 419–429.
  3. J. Devillers and A. T. Balaban, Topological indices and related descriptors in QSAR and QSPAR, CRC Press, 2000.
  4. S. Fajtlowicz. On conjectures of Graffti-II. Congr. Numer. 60 (1987) 187–197.
  5. B. Furtula, K. Ch. Das and I. Gutman, Comparative analysis of symmetric division deg index as potentially useful molecular descriptor, Int. J. Quantum Chem. 118 (17) (2018) e25659.
  6. B. Furtula, A. Graovac and D. Vukičecić, Augmented Zagreb index, J. Math. Chem. 48 (2) (2010) 370–380.
  7. C. K. Gupta, V. Lokesha, S. B. Shetty and P. S. Ranjini, Graph operations on symmetric division deg index of graphs, Palestine. J. Math. 6 (1) (2017) 280–286.
  8. C. K. Gupta, V. Lokesha, S. B. Shwetha and P. S. Ranjini, On the symmetric division deg index of graph, Southeast Asian Bull. Math. 40 (1) (2016) 59-80.
  9. I. Gutman. Degree-based topological indices, Croat. Chem. Acta 86 (4) (2013) 351–361.
  10. I. Gutman and K. Ch. Das, The firstzagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (1) (2004) 83–92.
  11. I. Gutman, B. Furtula, and C. Elphick, Three new/old vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 617–632.
  12. I. Gutman, E. Milovanović, and I. Milovanović. Beyond the Zagreb indices, AKCE Int. J. Graphs and Combin. 2 (2018) 307-312.
  13. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Science & Business Media, Berlin, Germany, 2012.
  14. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons, Chem. Phys Lett. 17 (4) (1972) 535–538.
  15. H. Hosoya, Topological index. a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Japan, 44 (9) (1971) 2332–2339.
  16. M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley-Interscience, New York, 2000.
  17. D. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1) (1995) 50–52.
  18. C. Liu, Y. Pan and J. Li, Tricyclic graphs with the minimum symmetric division deg index, Discrete Math. Lett. 3 (2020) 14–18.
  19. V. Lokesha and T. Deepika, Symmetric division deg index of tricyclic and tetracyclic graphs, Int. J. Sci. Eng. Res, 7 (5) (2016) 53–55.
  20. G. Mohanapriya and D. Vijayalakshmi, Symmetric division degree index and inverse sum index of transformation graph, J. Physics: Conf. Series 1139 (1) (2018) 012048. DOI: 10.1088/1742-6596/1139/1/012048.
  21. J. L. Palacios, New upper bounds for the symmetric division deg index of graphs, Discrete Math. Lett. 2 (2019) 52–56.
  22. Y. Pan and J. Li, Graphs that minimizing symmetric division index deg, MATCH Commun. Math. Comput. Chem. 82 (1) (2019) 43–55.
  23. M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (23) (1975) 6609–6615.
  24. K. Varmuza, M. Dehmer and D. Bonchev (Eds.), Statistical Modelling of Molecular Descriptors in QSAR/QSPR, Wiley−VCH, Weinheim, Germany, 2012.
  25. A. Vasilyev, Upper and lower bounds of symmetric division deg index, Iranian J. Math. Chem. 5 (2) (2014) 91–98.
  26. D. Vukičecić and M. Gašperov, Bond additive modeling 1. Adriatic indices, Croat. Chem. Acta 83 (3) (2010) 243–260.
  27. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1) (1947) 17–20.
Iranian Journal of Mathematical Chemistry
Volume 11, Issue 3
September 2020
Pages 141-159

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