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
(a)
(b)
(c)
(d)
Denote (see Figure 1(a)).
For , let be a graph obtained from by attaching a pendant vertex to one pendant vertex of except for (see Figure 1(d)). Then . Let be a graph obtained from by attaching pendant vertices to one pendant vertex of except for (see Figure 1(c)). Then
Yu et al. [10] characterized that had the minimum hyper-Wiener index of all trees on vertices with diameter . In the following, we firstly give possible trees with the second-minimum hyper-Wiener indices in the set .
Denote , , , and .
Lemma 8. Let . Then , or , or , or .
Proof. If , the conclusion is obvious. If , let be a path of length in with Let Since , we consider two cases.
Case 1 (). By Lemma 3, By Corollary 2, , where
So Case 2 (). Let and with , and Then as Let be subtrees of which contain , and By Lemma 3, we can obtain a tree obtained from by attaching pendant vertices to , respectively, such that . By Corollary 2, we can obtain a tree such that So , or .
Lemma 9. Let be a path of order . Then for Moreover, if , and ; if , and
Proof. By the definition of and , we have Then , and thus the results hold.
Lemma 10. (1) For any , with equality if and only if
(2) For any , with equality if and only if
(3) For any , with equality if and only if
(4) For any , with equality if and only if
Proof. Let and two paths in Lemma 6. We obtain the results of (1), (2), and (3). In the following, we prove (4).
Let be a tree which is as small as possible in the set . Let be a path of length in with , and be a pendant vertex of which is adjacent to . By Lemma 7 and the choice of , we know that Note that , and then by Corollary 5, we have Thus, if , ; if , , with equality if and only if or . So, for any , with equality if and only if
Lemma 11. Let , and then(1);(2)
Proof. (1) We denote , .
Note that , and then by Corollary 5, we have Namely, Note that , and then by Lemma 4, we have Namely,(2) By Lemma 9, , and By Corollary 5 and Lemma 9, Namely,
Theorem 12. Let Then with equality if and only if
Proof. If , note that contains no other trees than , and by Lemma 6, take and two paths , and we have
So, with equality if and only if
If , by Lemmas 8, 10, and 11, we have with equality if and only if
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 11071001, 11071002, 11072059, and 61272530, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Natural Science Foundation of Department of Education of Anhui Province of China under Grant nos. KJ2015ZD27 and AQKJ2015B010.