On discriminativity of Zagreb indices

Document Type: Research Paper

Author

University of Zagreb

Abstract

Zagreb indices belong to better known and better researched topological indices. We investigate here their ability to discriminate among benzenoid graphs and arrive at some quite unexpected conclusions. Along the way we establish tight (and sometimes sharp) lower and upper bounds on various classes of benzenoids.

Keywords


1. A. R. Ashrafi, T. Došlić, A. Hamzeh, The Zagreb coindices of graph operations,
Discrete Appl. Math. 158 (2010) 1571–1578.

2. J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors:
the equivalence of Zagreb indices and walk counts, MATCH Commun. Math.
Comput. Chem. 54 (2005), 163–176.
3. D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete
Math. 185 (1998) 245–248.
4. S. J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Lec. Notes
in Chemistry, Springer, Heidelberg, 1988.
5. K. Ch. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete
Math. 285 (2004) 57–66.
6. K. Ch. Das, I. Gutman, Some properties of the second Zagreb index, MATCH
Commun. Math. Comput. Chem. 52 (2004) 103–112.
7. T. Došlić, Vertex-Weighted Wiener Polynomials for Composite Graphs, Ars Math.
Contemp. 1 (2008) 66–80.
8. I. Gutman, K. Ch. Das, The first Zagreb index 30 years after, MATCH Commun.
Math. Comput. Chem. 50 (2004) 83–92.
9. M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The first and second Zagreb
indices of graph operations, Discrete Appl. Math. 157 (2009) 804–811.
10. M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, S.Wagner, Some new results on
distance-based graph invariants, Europ. J. Combin. 30 (2009) 1149–1163.
11. D. J. Klein, T. Došlić, D. Bonchev, Vertex-weightings for distance moments and
thorny graphs, Discrete Appl. Math. 155 (2007) 2294–2302.
12. V. Nikiforov, The sum of the squares of degrees: an overdue assignment,
arXiv:math/0608660.
13. S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb Indices 30 Years
After, Croat. Chem. Acta 76 (2003) 113–124.
14. D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River,
1996.
15. S. Yamaguchi, Estimating the Zagreb indices and the spectral radius of triangle-and
quadrangle-free connected graphs, Chem. Phys. Lett. 458 (2008) 396–398.
16. Y. S. Yoon, J. K. Kim, A relationship between bounds on the sum of squares of
degrees of graph, J. Appl. Math. & Comput. 21 (2006) 233–238.
17. B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices,
Chem. Phys. Lett. 394 (2004) 93–95.
18. B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of seriesparallel
graphs, Int. J. Quant. Chem. 107 (2007) 875–878.
19. B. Zhou, N. Trinajstić, On reciprocal molecular topological index, J. Math. Chem.
44 (2008) 235–243.