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Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 6079450, 5 pages
Research Article

Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

1School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
2Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, Pakistan
3Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan
4Department of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
5Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USA
6School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Correspondence should be addressed to Jia-Bao Liu

Received 19 September 2016; Revised 16 December 2016; Accepted 27 December 2016; Published 1 February 2017

Academic Editor: J. R. Torregrosa

Copyright © 2017 Wei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as , where is the degree of the vertex in a graph . In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs among -vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.