Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 6079450, 5 pages

https://doi.org/10.1155/2017/6079450

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

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

#### Abstract

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.

#### 1. Introduction

A topological index of a graph is a number with this property that, for every graph isomorphic to , Top() = Top(). In 1947, Wiener determined the most widely known topological descriptor, the Wiener index [1]. He used it to determine physical properties of paraffin. The Wiener index of a graph is equal to the sum of distances between all pairs of vertices of related graphs. Numerous indices have been explored. The Zagreb indices are the most important topological indices, introduced by Gutman and Trinajstić more than thirty years ago [2]. For a graph , the first and second Zagreb indices, and , respectively, are defined as In 1972, Zagreb indices first appeared in the topological formulas for the total -energy of conjugated molecules [2]. For applications in QSPR/QSAR, latest results are referred to [3–8].

In 2004, Miličević et al. [9] reformulated Zagreb indices in terms of edge-degrees instead of vertex-degree as follows: where is the degree of the edge in , defined by with , and means that the edges and are adjacent. Some results related to and are given in [10–12].

In 2013, Shirdel et al. [13] introduced a new degree-based topological index named hyper-Zagreb index as Recently, the multiplicative versions of Zagreb indices are studied well in [14]. Motivated by these results [15–18], we explore the properties for the hyper-Zagreb index.

Let be a simple and connected graph with vertex set and edge set . For a vertex , denotes the set of all neighbors of in . In a graph , the number of independent cycles is called its cyclomatic number and is equal to . Recall that graphs with are referred to as trees, unicyclic graphs, and bicyclic graphs, respectively. , , and denote the star, path, and cycle on vertices, respectively. Let , and then let be a subgraph of by deleting vertex and adjacent edges. For , let be a subgraph of by deleting an edge . Let be a nontrivial graph and let be its vertex. If is obtained from by fusing a tree at , then we say that is a subtree of and is its root. The fusions of two vertices and in are denoted by . In order to exhibit our results, we introduce some graph-theoretical notations and terminology. For other undefined ones, see the book [19].

In this paper, we characterize the extremal properties of the hyper-Zagreb index. In Section 2, we present some graph transformations which increase or decrease . In Section 3, we determine the extremal acyclic, unicyclic, and bicyclic graphs with maximum and minimum hyper-Zagreb index.

#### 2. Graph Transformations

In this section, we will introduce some graph transformations, which increase or decrease the hyper-Zagreb index. These transformations will help to prove our main results. The following one from to strictly decreases the hyper-Zagreb index of a graph.

*Transformation 1. *Let be a nontrivial connected graph and is a given vertex in . Let be a graph obtained from by attaching two paths: of length and of length . If , we say that is obtained from by Transformation 1, as shown in Figure 1.