Abstract
The Wiener index of a graph is defined as the sum of distances between all unordered pairs of its vertices. We found that finite steps of diameter-growing transformation relative to vertices can not always enable the Wiener index of a tree to increase sharply. In this paper, we provide a graph transformation named diameter-growing transformation relative to pendent edges, which increases Wiener index of a tree sharply after finite steps. Then, twenty-two trees are ordered by their Wiener indices, and these trees are proved to be the first twenty-two trees with the first up to sixteenth smallest Wiener indices.
1. Introduction
Graph theory has been applied in many areas of engineering extensively, such as mechanical design and manufacturing and chemical engineering. The link in the mechanism can be regarded as the vertex in the graph theory, kinematic pair can be regarded as an edge, and then the topological configuration of the mechanism is abstracted as the graph in graph theory. Therefore, relevant theories of graph theory can be used to analyse and discuss the nature and characteristics of the mechanism and especially be used to explore the movement patterns of the mechanism. Analogously in chemical engineering, a chemical molecule is usually regarded as a two-dimensional graph, and the graph’s vertices represent atoms and edges represent chemical bonds. Thus, the graph determines the topological properties of the given molecule. In fact, chemists are often interested in the Wiener indices of certain trees which represent certain molecular structures. In this paper, we study the trees with the first up to sixteenth smallest Wiener indices.
All graphs considered in this paper are simple and connected. Let be a graph that vertex set is and edge set is . For any , the distance between and in is the length of a shortest path. As the oldest graph-based structure descriptor (topological index), Wiener index was first provided by Wiener in [1] and defined as Its mathematical properties and chemical applications have been extensively studied in many literatures. In [1], Wiener provided an alternative calculating form of Wiener index for each tree ; i.e., where (resp., ) is the number of vertices of the component of containing (resp., ). It should be mentioned that the above equation is valid only for trees.
There have been many related achievements on Wiener index and they can be roughly divided into the following categories.
(i) Basic Theory and Calculation of Wiener Index (See [2–4]). Many known results about Wiener index of trees were introduced in [2], including the calculation of Wiener index, combinatorial expressions for Wiener index, and the relationship between Wiener index and some characteristic indicators of graph. Manuel et al. in [3] presented a method for calculating Wiener index of certain chemical graphs without using distance matrix. Graovac et al. provided a modification of Wiener index which considers the symmetry of a graph in [4].
(ii) The Wiener Index for Several Types of Trees (See [5–7]). Fuchs et al. in [5] studied the Wiener index for several types of random digital trees. Wagner in [6] demonstrated that every integer () is the Wiener index of a star-like tree. [7] derived formulas for calculating the Wiener indices when a tree was modified.
(iii) Other Topological Indices Associated with Wiener Index (See [8–16]). The Steiner Wiener index, its inverse problem and the Steiner hyper-Wiener index of a graph were studied in [8–11]. References [12–14] investigated the terminal Wiener index of graphs. For the results about the trees’ reverse Wiener indices and ordering problem by their reverse Wiener indices, one can refer to [15, 16].
Since every atom has a certain valency, chemists especially focus on the trees with the maximum (resp., minimum) Wiener index. References [17–22] characterized some -vertex trees which minimize or maximize the Wiener index over the set of trees with some degree and diameter restrictions.
It is natural to consider not only the maximum and minimum Wiener indices of given trees, but also the order of trees by Wiener indices. Ordering trees by their Wiener indices can help us to analyse and evaluate the relationship between the Wiener indices and the structures of chemical molecules represented by the certain trees. Actually, Dong et al. in [23] ordered fifteen trees with in terms of the number of nonpendent edges. The trees with given matching number have been ordered by their Wiener indices in [24].
Motivated by the results above, we extend the conclusion in [23] to general case in this paper. In Section 2, we present an example which shows that the result of Lemma 5 in [20] is not always true. As a by-product, we provide diameter-growing transformation relative to pendent edges; this transformation increases Wiener index sharply after finite steps. Then, we amend the proof of Theorem 2 in [20] by applying this transformation. In Section 3, we order twenty-two trees by their Wiener indices and prove that these trees on are the first twenty-two trees with the first to sixteenth smallest Wiener indices.
Throughout the paper, we denote the set of trees on vertices by , and the set of trees on vertices with diameter by .
2. The Trees’ Diameter-Growing Transformation Relative to Pendent Edges
The following example shows the conclusion of Lemma 5 in [20] that is not always true. In fact, finite steps of diameter-growing transformation of relative to vertices can not always enable to increase sharply.
Example 1. For each , the trees and are defined as in Figure 1. Then
Proof. From equality (2), it follows that and . So, it is obvious that holds for all .
To prove that Theorem 2 in [20] is still true, we firstly show that the Wiener index increases sharply after finite steps of diameter-growing transformation of trees relative to pendent edges.
Lemma 2. Let with the longest path . Suppose . Then . In addition, the procedure from to shown in Figure 2 is called a diameter-growing transformation of relative to the pendent edge .
Proof. Let be the pendent edge of , be the nonpendent edge of , and and be the two components of containing and . Then and . Thus, it follows that
In fact, from the proof of Lemma 2, we can get the diameter-growing transformation relative to pendent edges in such general case as follows.
Remark 3. For any tree defined as in Figure 3 and being a pendent edge of , it follows that .
Let be the set of trees of order with exactly vertices of maximum degree and be the tree shown in Figure 4.
Theorem 4 (Theorem 2 in [20]). Let , where . Then with equality if and only if .
Actually, the proof of Theorem 4 (Theorem 2 in [20]) is just needed to be modified by replacing Lemma 5 in [20] with Lemma 2 in our paper.
3. The First Twenty-Two Trees in with the First up to Sixteenth Smallest Wiener Indices
In this section, firstly, we give the order of twenty-two trees on vertices according to Wiener indices by using the diameter-growing transformation relative to pendent edges.
Theorem 5. For any , let , ) be some trees on vertices listed as in Figure 5, where is the diameter of each tree. Then we have
Proof. From equality (2), we can easily get All the above completes proof of the theorem.
For convenience, we denote
Next, we are going to show that holds for any with . The proof process is organized according to the diameters of trees.
Theorem 6. For each , let with . Then where tree is defined as in Figure 6.
Proof. On the one hand, from we get that for any . On the other hand, for any with and , is just shown as in Figure 7.
It is easy to see that holds for any with . Therefore, we get for any with and .
Next, we are going to prove that is always true for each with and .
Lemma 7. For any , , and , let be the tree defined as in Figure 8. Then holds for any .
Proof. For each , we have It follows that Then the proof will be divided into three cases as follows.
Case 1 (). For each , we have . Then, from for any , we get Meanwhile, we have , , and .
Case 2 (). For each , we have . From , it follows that Moreover, .
Case 3 (). For each , we have , which results in . Together with inequality (15), we get Since , then, for each and , from , it follows that All the above implies that holds for any .
Lemma 8. For each , , , and , let be the tree defined as in Figure 9. Then holds for any .
Proof. The proof will be divided into the following cases.
Case 1 (). Then , and for each . Using Lemma 7, it is obvious that holds for any .
Case 2 ()
Case 2.1 (). Then . We get So, it follows that However, .
For any with , this case can be divided into two subcases as follows.
Case 2.1.1 (). If , from and , it follows that , and hence Note that , so for each .
If , then . So . In fact, for each , from , we conclude that Thus, we have for any .
Case 2.1.2 (). For each , from , we can get . Then it follows that .
Case 2.2 ()
Case 2.2.1 (). Since , for any , then for some . So, we have for .
Actually, from and , we get . Combining with , it follows that Case 2.2.2 (). Then we have . Indeed, from and , we get that , and hence From all of the above cases, it turns out that holds for any .
Lemma 9. Let be a tree on vertices and be a tree transformed from just as shown in Figure 10. Then, for each given , holds for any with . It is equivalent to say that holds for , where is the order of .
Proof. Actually, for any . In addition, from , it follows that , if and only if .
Actually, the following theorem is a result concluded from Lemmas 7, 8, and 9.
Theorem 10. For each , and , let . Then
Proof. The proof will be divided into four cases as follows.
For each given , let be the set of all branching vertices of (where is the degree of vertex in ). From the assumption that and , it is obvious that .
Case 1 (). Being from , together with Lemmas 8 and 9, we get .
Case 2 (). The assumption that and implies that has at least two nonpendent vertices. If has exactly two nonpendent vertices, from Lemmas 7 and 9, together with , it follows that . Otherwise, from Lemmas 8 and 9, together with , it follows that .
Case 3 (). The assumption that and implies that has at least two nonpendent vertices. If has exactly two nonpendent vertices, from Lemmas 7 and 9, together with , it follows that . Otherwise, from Lemmas 8 and 9, it follows that .
Case 4 (). From Lemmas 8 and 9, we get that .
Theorem 10 demonstrated the Wiener indices of the trees with diameter 4. Next, we will study the Wiener indices of the trees with diameters .
It is well-known that [25] obtained the trees with minimum and second-minimum Wiener indices among all the trees with vertices and diameter . To be more precise, the authors in [25] pointed out that the tree which attains the minimum Wiener index in is and that the tree which attains the second-minimum Wiener index in is . The trees and are shown as in Figure 11.
Lemma 11 (Theorem 3.6 in [25]). Let be the set of all trees on order with diameter .
(i) For any with , the tree which attains the minimum Wiener index in the set is .
(ii) For being odd and , the tree which attains the second-minimum Wiener index in the set is .
(iii) For being even and , the tree which attains the second-minimum Wiener index in the set is or .
Lemma 12. For any , we have
Proof. In fact, holds if is even. Meanwhile, holds if is odd. From Remark 3, it follows that Thus, for any tree in Figure 12 and any in Figures 13 or 14, we get for each with whenever is even or odd.
From Lemmas 11 and 12, it is easy to get the following conclusion.
Theorem 13. Let with . Then .
Proof. Since we get for any . From Lemma 12, it follows that Moreover, from Lemma 11, we get for any tree on vertices with diameter . By using Lemma 11 again, we get for any tree on vertices with diameter , except for and . Thus, we conclude that holds for any tree on vertices with diameter .
Theorem 14. For each , let . Then
Proof. By combining the previous arguments of Theorems 5, 6, 10, and 13, the proof is completed.
4. Conclusion
In this paper, we provide a transformation of trees named diameter-growing transformation relative to pendent edges. By using this transformation, we amend the proof of Theorem 2 in [20], then give the order of twenty-two trees by their Wiener indices, and prove that these trees are the first twenty-two trees with the first up to sixteenth smallest Wiener indices.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the Natural Science Foundation of Shanxi Province (no. 201601D011003).