ISRN Applied MathematicsVolume 2011 (2011), Article ID 421456, 9 pagesdoi:10.5402/2011/421456

Research Article

On the Maximal Eccentric Distance Sums of Graphs

Jianbin Zhang¹ and Jianping Li²

¹School of Mathematics, South China Normal University, Guangzhou 510631, China²Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510090, China

Received 12 March 2011; Accepted 14 April 2011

Academic Editor: H. Akçay

Copyright © 2011 Jianbin Zhang and Jianping Li. Jianbin Zhang and . 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.

If is a simple connected graph with vertex , then the eccentric distance sum of , denoted by , is defined as , where is the eccentricity of the vertex and is the sum of all distances from the vertex . Let . We determine the -vertex trees with, respectively, the maximum, second-maximum, third-maximum, and fourth-maximum eccentric distance sums. We also characterize the extremal unicyclic graphs on vertices with respectively, the maximal, second maximal, and third maximal eccentric distance sums.

1. Introduction

Let be a simple connected graph with vertex set and edge set . Let be the distance (or length of a shortest path) between vertices and in . For a vertex , the eccentricity is the maximum distance from to any other vertex, and is the sum of all distances from . The eccentric distance sum of (EDS) is defined as

This graph invariant was proposed by Gupta et al. in [1]. Like the Wiener index [2–4] and eccentric connectivity index [5–9], it turned to have high discriminating power and excellent predictability both with regard to biological and physical properties and to provide valuable leads for the development of safe and potent therapeutic agents of diverse nature. So, it is of interest to study the mathematical properties of this invariant.

Yu et al. [10] considered the -vertex trees and unicyclic graphs with minimal eccentric distance sums, respectively. Ilić et al. [11] proved that path is the unique extremal trees with vertices having maximum eccentric distance sum, and provided various lower and upper bounds for the eccentric distance sum. In this paper, we determine the -vertex trees with, respectively, the maximum, second-maximum, third-maximum, and fourth-maximum eccentric distance sums for . We also characterize the extremal unicyclic graphs with the maximal, second maximal, and third maximal eccentric distance sums.

2. The Trees with Maximal Eccentric Distance Sums

Lemma 2.1. Let be a vertex of a connected graph with at least two vertices. Let be the graph obtained by identifying u and a vertex of a path , where . Then, .

Proof. By the definition of EDS, we have Denoted by , , and respectively, the three sums of right equality above, we only need to prove the following three inequalities: , , .Note that for any , and and thus, the inequality holds.Note that for any , , , and thus . So, it suffices to prove .Let and , and we distinguish the following two cases:Case 1 (). By direct calculation, it follows that For , it is easily seen that and . Furthermore, , and thus, .Case 2 (). For , it is easily seen that , . Then, where the last inequality follows as for .

By Lemma 2.1 the inequality follows easily. Note that has the number of pendent vertices greater than that of . Let be a tree with at least pendent vertices, where , by applying the above transformation to repeatedly, then we can obtain a new tree with exactly pendent vertices, which has larger eccentric distance sum. So it is easy to prove the following result, which is also obtained by Ilić et al. in [11].

Theorem 2.2. Among all trees with vertices, has the maximal eccentric distance sum.

Let be the tree obtained from by attaching a pendent vertex to , where .

Lemma 2.3. Let . Then,

Proof. Suppose that has the same vertex labeling as the above definition. We have two cases based on the parity of . If is even, then If is odd, then we make a similar calculation as above and obtain that These complete the proof.

Lemma 2.4. Among all trees on vertices, where , has the maximal eccentric distance sum except , , and .

Proof. suppose that is an -vertex tree different from , , and . Let be the number of pendent vertices of , then .If , then is is a tree obtained by identifying three pendent vertices of three paths. Denote by , and , respectively, the lengths of the three paths, where and . Here, we denote it by . Clearly, . Suppose first that . Then, with . For , by Lemma 2.1, we have , and thus, with equality if and only if . Now suppose that , then for , and by Lemma 2.1, we can obtain a new tree or , which is not isomorphic to , , and and has larger eccentric distance sum.If by applying transformation of Lemma 2.1 to repeatedly, we can obtain a new tree with exactly four pendant vertices and larger eccentric distance sum. Thus, it suffices to consider the case .Now, suppose that . In this case, has at most two vertices with degree more than 2.Case 1. If has exactly two vertices with degree more than 2, say and , then . Suppose that the length of path connecting and is , the lengths of pendent paths at are , and the lengths of pendent paths at are , . We denote this tree by , where , , and . If , then , and by Lemma 2.1 and above proof, we have , where we suppose that . If , then by Lemma 2.1 and the result above we have . If , or , . Applying similar proof of Lemma 2.3, it is easily proven that , , and thus, we only need to prove that .Here, we write instead of . Then, is a tree obtained from the path by joining an isolated vertex to and an isolated vertex to . Let be a tree obtained from by deleting the edge and adding the edge . If is odd, then for , ; for , It follows that Thus, .Similarly, for even , we have and the inequality also holds.Case 2. If has an unique vertex of degree greater than 2, say , then , and is a tree obtained by identifying four pendent vertices of four paths. Denote by this tree, where are the lengths of the four pendent paths respectively and , . If , by Lemma 2.1, we have . Similarly, we can show the result for and . In the following, we will prove that . We label the vertices of and such that can be viewed as obtained from the path by joining three isolated vertices , and to , and can be viewed as obtained from the path by joining an isolated vertex to . Clearly, for , for and , and . Note that then and this completes the proof.

From Lemmas 2.1 and 2.4, We have the following

Theorem 2.5. If , then , , and are the unique trees with the second-maximal, third-maximal, and fourth-maximal eccentric distance sums among the trees on vertices.

3. The Unicyclic Graphs with Maximal Eccentric Distance Sums

Let be the graph obtained from a path by joining the vertex to and . Let be the graph obtained from a path by joining the vertex to and , and be obtained by joining to and .

Theorem 3.1. Let be a graph with vertices and edges; that is, is an unicyclic graph, where . Then, , with equality if and only if .

Proof. If and , then we can always find an edge of such that is a tree with at least three pendent vertices and , . It follows from Lemma 2.3 that Suppose that and is any vertex of . If is even, then By the definition of EDS, we have It is easily checked that and for , , , , , . Then, We can prove the result for odd number similarly, and thus complete the proof.

Theorem 3.2. Let . Then, , are the graph with, respectively, the second-maximal and third-maximal eccentric distance sums among all unicyclic graphs on vertices.

Proof. If has a spanning tree such that , then Now suppose that any spanning tree of is one of . Then, must be isomorphic to , or . It can be proven easily that by similar proof of Theorem 3.1. And by directed computation, we obtain that . Thus, the result follows.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11001089), and by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant no. (2009) 400). The authors thank the referees for valuable comments and suggestions.

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