Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 7241349, 5 pages

http://dx.doi.org/10.1155/2016/7241349

## The Hyper-Wiener Index of Trees of Order with Diameter

^{1}School of Mathematics & Computation Sciences, Anqing Normal University, Anqing 246011, China^{2}Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China^{3}Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia^{4}Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia^{5}Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 23 March 2016; Accepted 31 August 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Gaixiang Cai 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-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener index is defined as with the summation going over all pairs of vertices in , and denotes the distance of the two vertices and in the graph . In this paper, we obtain the second-minimum hyper-Wiener indices among all the trees with vertices and diameter and characterize the corresponding extremal graphs.

#### 1. Introduction

Let be a simple graph of order with vertex set and edge set . The distance between two vertices of , denoted by or , is defined as the minimum length of the paths between and in . The diameter of a graph is the maximum distance between any two vertices of .

The Wiener index of a graph , denoted by , is one of the oldest topological indexes, which was first introduced by Wiener [1] in 1947. It is defined as where the summation goes over all pairs of vertices of . The hyper-Wiener index of acyclic graphs was introduced by Randić in 1993 [2]. Then Klein et al. [3] extended the definition for all connected graphs, as a generalization of the Wiener index. Similar to the symbol for the Wiener index, the hyper-Wiener index is traditionally denoted by . The hyper-Wiener index of a graph is defined as

Let Then We denote , and then

Recently, the properties and uses of the hyper-Wiener index have received a lot of attention. Feng and Ilić [4] studied hyper-Wiener indices of graphs with given matching number. Feng et al. [5] researched the hyper-Wiener index of unicyclic graphs. Feng et al. [6] discussed the hyper-Wiener index of bicyclic graphs. Gutman [7] obtained the relation between hyper-Wiener and Wiener index. Xu and Trinajstić [8] discussed hyper-Wiener index of graphs with cut edges. M.-H. Liu and B. L. Liu [9] determined trees with the seven smallest and fifteen greatest hyper-Wiener indices. Yu et al. [10] studied the hyper-Wiener index of trees with given parameters.

All graphs considered in this paper are finite and simple. Let be a simple graph with vertex set and edge set . For a vertex , the degree and the neighborhood of are denoted by and (or written as and for short). A vertex of degree 1 is called pendant vertex. An edge incident with the pendant vertex is called a pendant edge. For a subset of , let be the subgraph of obtained from by deleting the vertices of and the edges incident with them. Similarly, for a subset of , we denote by the subgraph of obtained from by deleting the edges of . If and , the subgraphs and will be written as and for short, respectively. For any two nonadjacent vertices and in graph , we use to denote the graph obtained from by adding a new edge . Denote by and the star and the path on vertices, respectively.

A tree is a connected acyclic graph. Let be a tree of order with diameter . If , then ; and if , then . Therefore, in the following, we assume that Let is a tree with order and diameter , Liu and Pan [11] characterized the minimum and second-minimum Wiener indices of trees in the set , and Yu et al. [10] characterized the minimum hyper-Wiener index of all trees on vertices with diameter . Motivated by these articles, we will characterize the second-minimum hyper-Wiener indices of trees in the set in this paper.

#### 2. Preliminaries

Lemma 1 (see [8]). *Let , , and be three connected graphs disjoint in pair. Suppose that are two vertices of , is a vertex of , and is a vertex of . Let be the graph obtained from , , and by identifying with and with , respectively. Let be the graph obtained from , and by identifying three vertices , and , and let be the graph obtained from , and by identifying three vertices , and . Then one has *

By Lemma 1, we have the following result.

Corollary 2. *Let be a graph and Suppose that is the graph obtained from by attaching pendant vertices to , respectively. Then or *

Let be two connected graphs with . Let be a graph with as its vertex set and as its edge set. We have the following result.

Lemma 3 (see [8]). *Let be a connected graph, be a tree of order , and Then and equality holds if and only if , where is the center of star *

Lemma 4 (see [6]). *Let be a connected graph of order , be a pendant vertex of , and . Then*(1)*;*(2)

*By Lemma 4 and the definition of hyper-Wiener index, we have the following result.*

*Corollary 5. Let be a connected graph of order , be a pendant vertex of , and . Then *

*Lemma 6 (see [9]). Let be a connected graph on vertices and be a vertex of . Let be the graph obtained from by attaching two new paths and of length and at , respectively, where and are distinct new vertices. Let . If , then *

*Let be a vertex of graph , and are distinct new vertices (not in ). Let be the graph , where is a new path Let and , where .*

*Lemma 7 (see [9]). Suppose is a connected graph on vertices or has only one vertex. If , then , where *

*3. Main Results*

*In this section, we will give the second-minimum hyper-Wiener index in the set . In order to formulate our results, we need to define some trees as follows.*

*Let be a tree of order obtained from a path by attaching pendant vertices to , respectively, where Denote (see Figure 1(b)), and we note that *