Extremal trees with respect to some versions of Zagreb indices via majorization

Document Type: Research Paper

Authors

1 Department of Mathematics, Khansar Faculty of Computer and Mathematical Sciences, Khansar Iran

2 Department of Mathematics, Khansar Faculty of Computer and Mathematical Sciences, Khansar Iran

Abstract

The aim of this paper is using the majorization technique to identify
the classes of trees with extermal (minimal or maximal) value of some topological
indices, among all trees of order n ≥ 12

Keywords

Main Subjects


1. A. T. Balaban, I. Motoc, D. Bonchev and O. Mekenyan, Topological indices for
structure activity correlations, Topics Curr. Chem.114 (1983) 21–55.
2. C. Bey, An upper bound on the sum of squares of degrees in a hypergraph, Discrete
Math. 269 (2003) 259–263.
3. K. C. Das, Sharp bounds for the sum of the squares of the degrees of a graph,
Kragujevac J. Math. 25 (2003) 31–49.
4. K. C. Das, A. Yurttas, M. Togan, A. S. Cevik and I. N. Cangu, The multiplicative
Zagreb indices of graph operations, J. Inequal. Appl. 90 (2013) 1–14.
5. D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete
Math. 185 (1998) 245–248.

6. J. Devillers and A. T. Balaban, Topological Indices and Related Descriptors in
QSAR and QSPR, Gordon and Breach Science Publishers (1999).
7. M. Eliasi, A simple approach to orther the multiplicative Zagreb indices of
connectedgraphs, Trans. Comb. 1 (2012) 17–24.
8. M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb
index, MATCH Commun. Math. Comput. Chem. 68 (2012) 217–230.
9. M. Eliasi and D. Vukicević, Comparing the multiplicative Zagreb indices, MATCH
Commun. Math. Comput. Chem. 69 (2013) 765–773.
10. M. Eliasi and A. Ghalavand, Ordering of trees by multiplicative second Zagreb
index, Trans. Comb. 5 (1) (2016) 49–55.
11. I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virt. Inst. 1
(2011) 13–19.
12. I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun.
Math. Comput. Chem. 50 (2004) 83–92.
13. I. Gutman, Graphs with smallest sum of squares of vertex degree, Kragujevac J.
Math. 25 (2003) 51–54.
14. I. Gutman and N. Trinajstić, Graph theory and molecular orbital.Total φ-electron
energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
15. M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley, New York (2000).
16. X. Li and J. Zheng, A unified approach to the extremal trees for different indices,
MATCH Commun. Math. Comput. Chem. 54 (2005) 195–208.
17. X. Li and H. Zhao, Trees with the first three smallest and largest generalized
topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004) 57–62.
18. M. Liu and B. Liu, Some properties of the first general Zagreb index, Australas. J.
Combin. 47 (2010) 285–294.
19. J. Liu and Q. Zhang, Sharp upper bounds for multiplicative Zagreb indices,
MATCH Commun. Math. Comput. Chem. 68 (2012) 231–240.
20. S. Nikolić, G. Kovacević, A. Miličević and N. Trinajstić, The Zagreb indices 30
years after, Croat. Chem. Acta 76 (2003) 113–124.
21. T. Rseti and I. Gutman, Relation between ordinary and multiplicative Zagreb
indices, Bull. Int. Math. Virt. Inst. 2 (2012) 133–140.
22. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley–VCH
(2000).
23. R. Todeschini and V. Consonni, New local vertex invariants and molecular
descriptors based on functions of the vertex degrees, MATCH Commun. Math.
Comput. Chem. 64 (2010) 359–372.
24. K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices
for trees, unicyclic and bicyclic raphs, MATCH Commun. Math. Comput. Chem. 68
(2012) 241–256.