#### Abstract

The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius two. In addition, we pose a conjecture concerning the minimum Wiener index of graphs with given radius. If this conjecture is true, it will be able to answer an open question by You and Liu (2011).

#### 1. Introduction

Let be a connected graph. For two vertices , the distance between and in is the length of the shortest path connecting and in . The eccentricity of a vertex in is the largest distance from to another vertex of ; that is, . The diameter of is the maximum eccentricity in , denoted by . Similarly, the radius of is the minimum eccentricity in , denoted by . The Wiener index or total  distance of , denoted by , is the sum of all distances between unordered pairs of distinct vertices of . In other words,

The average distance of is defined as . A vertex is called a center of if . It is well known that every tree has either exactly one center or two adjacent centers. For the results on Wiener index of trees, we refer to Dobrynin et al. [1], and for some more results on Wiener index of graphs, one can see [212].

Plesník [13] addresses a problem on distance of graphs, which remains unresolved.

Problem 1. What is the maximum total or average distance among all graphs of order with diameter ?

To see how hard to solve Problem 1, one just consider the following conjecture, due to DeLaViña and Waller [14].

Conjecture 1 (see [14]). Let G be a graph with diameter and order 2d + 1. Then , where denotes the cycle of length .

For any connected graph , . By considering the close relationship between the diameter and the radius of a graph, it is natural to consider the following problem.

Problem 2. What is the maximum total or average distance among all graphs of order with radius ?

By Lemma 3 in Section 2, Problem 2 is more tractable than Problem 1; the graph with maximum total distance among all graphs of order with radius must be a tree. But, Problem 2 still seems to be quite challenging. The main focus of this paper is to tackle Problem 2 when the case . We will show that the graph with the maximum Wiener index among all connected graphs with radius two is a tree with somewhat fractal-like structure when the number of vertices of graphs goes to infinity. It provides some clues for the further investigation of Problem 2 for the case .

The following problem remains open, see You and Liu [10].

Problem 3. What is the minimum total or average distance among all graphs of order with radius ?

It is easy to see that the complete graph has the minimum Wiener index among all connected graphs of order (and so among all graphs of order with radius one). For the case of radius two, it is not hard to prove that the extremal graph of Problem 3 should be the following: , if is even, where is a perfect matching; , if is odd, where is a maximum matching of and is an edge incident with the vertex of degree in .

We denote by the graph obtained in the following way, where and are three positive integers with , , and : let , and be four consecutive vertices on ; replace with a clique of order , replace with a clique of order , join each vertex of a clique to all vertices of the other clique, join to the all vertices of the , and join to all vertices of . It is clear that is a graph of order with radius , and for any .

Conjecture 2. Let and be two positive integers with and . For any graph of order with radius , , with equality if and only if for .

#### 2. Definitions and Lemmas

Lemma 3. For any connected graph , there exists a spanning tree of , such that and .

Proof. Let be a center of and be a BFS-tree of rooted at . The tree is one, as we desired.

The path number of a graph is defined to be the maximum cardinality of a subset of vertices that induces a path. Erdős et al. [15] showed that for any connected graph , .

Lemma 4. For a tree with radius , .

Proof. Let be the longest path of . Since is a tree, . It is clear that a center of must be sitting on , and . The result then follows.

Lemma 5. Let be a tree of order with the maximum Wiener index among all trees with radius two. If , then , and , otherwise.

Proof. If , the result is trivial. So, let . By Lemma 4, . By contradiction, suppose that , and let be a path on four vertices in . Every other vertex not on must be adjacent to or . Without loss of generality, let , and let be a pendent vertex adjacent to . Set . It is easy to check that , contradicting the choice of .

Lemma 6. Let be a tree. Then(1)the center of is a vertex or two adjacent vertices,(2)the center of is one vertex if and only if .

The above two lemmas assert that if is a tree of order with the maximum Wiener index among all trees of order at least five with radius two, then has the unique center. We denote by the tree of order with radius two and unique center such that , , and for each . In particular, , with for any .

Lemma 7. Let be the tree as described above. Then

Proof. Let . Hence , where By simple calculation, .

The trees and when are illustrated in Figure 1. We will see that if in the next corollary. Note that if , each vertex in , not a pendent vertex, has children. In this sense, has a fractal-like structure. Our main goal is to prove that if is a tree of order with the maximum Wiener index among all trees with radius two, then , where can be determined. It will be shown in Theorem 9. For smaller value of , one can see that those extremal graphs with radius two in Figure 2, and there exists two extremal graphs when for some ; for instance, see the graphs for () and () in Figure 2.

Corollary 8. Let and be two positive integers such that and for some integer , and let . Then and if and , then .

Proof. By the notation of the previous lemma, In , the number of equal to is and the number of equal to is . So, by Lemma 7, If and , by the above formula, Moreover, since and
this proves .

#### 3. Main Results

For a clear understanding of our main theorem as follows, we make a remark. If for some positive integer , then . By Corollary 8, .

Theorem 9. Let be a tree with the maximum Wiener index among all trees of order with radius two. If for some positive integer , then , and otherwise, where .

Proof. The result is trivial for , and assume that . By Lemma 5, let be the unique center of , and let . Set and for . Lemma 7 implies that for any , and thus for some . To show , as specified in the theorem, it suffices to show that for any .
By Corollary 8, , where , , we have
Define a function , which is an upper bound of . Its derivative has the unique root . Observe that and since ,
By the definition of , , where and , and moreover, if , then ; if , then .
Case  1 (). Since
Moreover, since and , we have
Thus, , and together with (11), we have for any . Observe from the above equation that for some if and only if . But, this contradicts the fact that if , then . It follows that for any .
Case  2 (). Since for , . Since ), we have Recall that and . Since and the fact that if , then , we have with equality only if . It follows that , with if and only if and . The proof is completed.

#### Acknowledgments

The research was supported by KPCME (no. 210243), NSFC (no. 11161046), and the Tianyuan Special Fund of NSFC (no. 11126113).