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Mathematical Problems in Engineering
Volume 2013, Article ID 524162, 9 pages
http://dx.doi.org/10.1155/2013/524162
Research Article

Properties and Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

College of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

Received 26 November 2012; Revised 26 February 2013; Accepted 26 February 2013

Academic Editor: Yong-Kui Chang

Copyright © 2013 Haizhou Song and Qiufen Wang. 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

We study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and show some applications. We obtain some results on the Laplacian spectral radius of a graph by grafting and adding edges. We also determine the structure of the maximal Laplacian spectrum tree among trees with vertices and pendant vertices (, fixed), and the upper bound of the Laplacian spectral radius of some trees.

1. Introduction

The theory of graph spectra has been established in the 1950s and 1960s. Chung has taken the investigation of the theory of graph spectra to a new level by a 45-minute report [1] presented at the World Congress of Mathematicians in 1994, and in monographs [2]. Applications of the theory of graph spectra have also been found in the fields of electrical networks and vibration theory [3, 4]. The wide range of the applications of the theory of graph spectra has led it to become a very active field of research of graph theory over the last thirty to forty years, and large numbers of results are continuously emerging.

There are many results on the (adjacency) spectral radius for different classes of graphs. Guo et al. [5] have studied the largest and the second largest spectral radius of trees with vertices and diameter . Guo and Shao [6] have studied the first (where represents the maximal integer not more than ) spectral radii of graphs with vertices and diameter . Wu et al. [7] have studied the spectral radius of trees on pendant vertices.

There are also many results on the Laplacian spectral radius for different classes of graphs. Hong and Zhang [8] have studied the upper and lower bounds for the Laplacian spectral radius of trees. Guo [9] has studied the second largest Laplacian eigenvalues of trees. In the paper, we further study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and obtain some applications on the Laplacian spectral radius of trees with pendant vertices.

Let be a simple connected graph with vertex set and edge set . Let denote the degree of vertex () of graph sometimes for convenience we denote by . If is the diagonal matrix of the vertex degree and is the adjacency matrix of , then the matrix is called the Laplacian matrix of graph . When is connected, is irreducible, and, by the Perron-Frobenius theorem [1012], the spectral radius is simple and there is a unique positive unit eigenvector. We refer to such an eigenvector as the Perron vector of . It is easy to see that is a positive semidefinite matrix and its rows sum to 0,   is singular. Hence, the eigenvalues of can be ordered as . The largest eigenvalue of is called the Laplacian spectral radius of the graph , denoted by . This has been used extensively in fields such as theoretical chemistry, combinatorial optimization, and communication networks [13, 14].

A pendant vertex of is a vertex of degree 1. Let denote the adjacent vertex set of in and the degree of . denotes the set of trees for vertices and pendant vertices (, fixed). The tree is obtained from a star and paths of almost equal length by joining each pendant vertex of to an end of one path. Obviously, . Assume ,   ; is shown in Figure 1.

524162.fig.001
Figure 1

In general, , , and , a tree is obtained from two stars and by joining their central vertices. For example, , , and is a path of length 6 (Figure 2).

524162.fig.002
Figure 2

In Section 2, we describe some properties of the eigenvector corresponding to the Laplacian spectral radius of a graph. Section 3 presents some applications of the eigenvector corresponding to the Laplacian spectral radius of a graph, including some results on the Laplacian spectral radius of a graph by grafting and adding edges. Then we obtain the structure of the maximal Laplacian spectrum tree among trees with vertices and pendant vertices (, fixed), and the upper bound of the Laplacian spectral radius for some trees.

2. Properties of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

For convenience, we denote by sometimes. Since the Laplacian matrix is a real symmetric matrix, we have the following theorem.

Theorem 1. Let be a simple connected graph of vertices. is the Laplacian matrix of , with vector . Then (1), where is the eigenvalue of ; (2); (3)if and , then .

Proof. It is proved by the definition.
Since is a real symmetric matrix, there exists an orthogonal matrix for which , where . Let , so .
is an orthogonal matrix and . Then, when , we have . Assume, without loss of generality, that . Thus,
Let , so we have , and then the above equality holds. Therefore, .
Since is a real symmetric matrix, there exists an orthogonal matrix for which , where , and is the eigenvector corresponding to of . Assume, without loss of generality, that .
Let , and then . Since , we have . Thus, and then . However, is the eigenvector corresponding to of , that is, . Thus, .

Definition 2. Let be a simple connected graph, . If vector satisfies and , then we call the standard Laplacian spectral vector of .

Theorem 3. Let be a tree, and . Let be the Laplacian matrix of and be the standard Laplacian spectral vector of . Then(1) is a real vector; (2).

Proof. Since is a real symmetric matrix, we have . Thus, is a real vector.
Suppose that is not established. Then there exists vertex set , which means ), where corresponds to vertex . Since is the standard Laplacian spectral vector of , we have . Thus, and there exists vertex that makes and vertex that makes .
Let form a root tree with as its root vertex. . Let be the subtree of obtained from vertex and all its descendants. Let , , when , and let be the vertex of subtree , when and be the vertex of subtree ).
Thus, we have , and . By Definition 2, we know that is the standard Laplacian spectral vector of . Then . We have . Thus, .
However, and there exists that makes , . Therefore, . This contradiction completes the proof.

Theorem 4. Let be a vertex of tree and be the pendant vertices of that are adjacent to . If is the standard Laplacian spectral vector of , where corresponds to the vertex . Then , .

Proof. is the standard Laplacian spectral vector of . By Definition 2 and Theorem 1, we have . , so , and then we have . Therefore, we have , . Thus, , .

Theorem 5. Let be the standard Laplacian spectral vector of tree , where corresponds to vertex , . Then (1)for , we have ; (2).

Proof. Since is the standard Laplacian spectral vector of , by Definition 2 we have . By Theorem 3, we have and . Thus, .
Suppose that there exists edge such that . Let tree form a root tree with as its root vertex. Assume, without loss of generality, that vertex is the parent node of the vertex (Figure 3). Let be the subtree of obtained from vertex and all its descendants, and let be a -layer tree, (Figure 4). Without loss of generality, we assume that , so .
Let (), , and ( are the vertices in the second layer of ), ( are the vertices in the th layer of ). Then and we have Thus, there exists a unit vector such that . This implies a contradiction with Theorem 1. Therefore, .
Since is the standard Laplacian spectral vector of , we have Then Thus, we have Therefore, . So .

524162.fig.003
Figure 3
524162.fig.004
Figure 4

3. Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius

In this section, we present some applications of the eigenvector of the Laplacian spectral radius of a graph.

Theorem 6. Let and be the vertices of tree T. Denote the distance of vertices and by and , where is an odd number and . Let be the graph obtained from by adding edge . Then .

Proof. Let be the standard Laplacian spectral vector of tree , where corresponds to vertex . Since is an odd number and , by Theorem 5 we have . Thus, Therefore, .

According to the well-known Courant-Weyl inequalities, we have the following lemma.

Lemma 7 (see [15]). Let be a graph with vertices and let be the graph obtained by inserting a new edge into . Let and be the eigenvalues of the Laplacian matrix of and , respectively. Then .

Lemma 8 (see [16]). Let be a connected graph for vertices with at least one edge. Then , where is the largest degree of the graph , and equality holds if and only if .

Theorem 9. Let be a connected graph for vertices, and let be the vertex of and be the pendant vertices of adjacent to vertex . Let be the graph obtained from by adding edges, where edges are among . Then .

Proof. If , by Lemma 8 we have . However, it is clear that , so . We suppose that .
Let be the graph obtained from by adding ) edges, where edges are among ). By Lemma 7, we have . To obtain the result, we only need to prove that .
Let be the standard Laplacian spectral vector of , then . For , we have Then, for and , we have since . Thus,
For , by Lemma 8 we have . So , , . Thus, we have That is, . This completes the proof.

Theorem 10. Let , be two vertices of tree . Suppose that are some vertices of , and is the standard Laplacian spectral vector of , where corresponds to vertex . Let be the tree obtained from by deleting the edges and adding the edges (Figure 5). If , then .

524162.fig.005
Figure 5

Proof. For tree , let form a root tree with vertex as the parent node of vertex ), . Let be the subtree of obtained from the parent node and and the descendants, descendants,…, descendants, . For tree ,   let form a root tree with vertex as the parent node of vertex ), . Let be the subtree of obtained from the parent node and and the descendants, descendants,…, and descendants. Let subtree be a -layer tree with vertex as the root node. Consider .
Let , , ( are the vertices in the second layer of ), ( are the vertices in the third layer of ),…, and ( are the vertices in the th layer of ). Then we have and That is, . Thus, Therefore, .
If , then the equalities in (12) hold. Thus, . By Theorem 1, we have . Thus, where is the degree of vertex of tree .
In addition, from we have where is the degree of vertex of tree .
By (13) and (14), we have Assume, without loss of generality, that .
If , by Theorem 5, we have which implies the contradiction .
If , by Theorem 5, we have which implies the contradiction .
Thus, if , then .

Corollary 11. Suppose that is as defined as in Theorem 10   is the standard Laplacian spectral vector of , and then .

Proof. Otherwise, if , by Theorem 10, we have . This contradiction completes the proof.

Corollary 12. Let be a simple connected graph with vertices. Let , , and be three distinct vertices, where and . Let be the simple connected graph obtained from by deleting the edge and adding the edge . Suppose that is the standard Laplacian spectral vector of , where corresponds to the vertex . is the standard Laplacian spectral vector of , where corresponds to the vertex . If , then .

Proof. Obviously, is obtained from by deleting the edge and adding the edge . Assume that . Since is a simple connected graph, by Theorem 10 we have . If , by Theorem 10 we have . This contradiction completes the proof.

Lemma 13 (see [17]). Let be a vertex of tree , and for positive integers and let denote the tree obtained from by adding pendant paths of length and at (Figure 6). If , then .

524162.fig.006
Figure 6

Theorem 14 (see [7, 8]). Among all the trees for vertices and pendant vertices, the maximal Laplacian spectral radius is obtained uniquely at .

Proof. We have to prove that if , then with equality only when . Let be the cardinality of the vertices of degree 3 or greater.
Case  1. . In this case, is a path of length , and hence . We have .
Case  2. . The result follows from repeated use of Lemma 13, and equality holds if and only if .
Case  3. . Let be the standard Laplacian spectral vector of , where corresponds to the vertex (). Suppose that are two vertices of of degree 3 or greater, and . Since is a tree, there is a unique path between and , and only one of the neighbors of , say , is on the path. Assume . Delete the edges (,) and add the edges () (). Then we obtain a new tree . Obviously, still has pendant vertices. By Theorem 10, we have and the cardinality of the vertices of degree 3 or greater decreases to .
If , repeat the above step to until the cardinality is only 1. Thus, we obtain trees and . Moreover, each has pendant vertices. Referring to Case 2, we have . Therefore, .
The cases complete the proof.

Theorem 15 (see [8]). .

Proof. The proof is obvious from Lemma 13.

For the sake of clarity, we identify a graph by its characteristic polynomial. Let be the characteristic polynomial of graph . For convenience, we denote by sometimes. First, we require some lemmas.

Lemma 16 (see [9]). Let be a cut edge of the simple connected graph . and are two connected branches of , where , . Then . Here, denotes the determinant obtained by deleting the row and column for vertex of from the determinant for , and denotes the determinant obtained by deleting the row and column for vertex of from the determinant for .

Lemma 17 (see [18]). Let be a path with vertices, let be the path obtained by adding a loop on a pendant vertex in , and let be the path obtained by adding loops on the two pendant vertices in . Suppose that the degree of contribution corresponding to each loop is 1 and , , . Then (1), (2), (3), ,(4), where and .

From Lemma 16, by induction we can obtain the following lemma.

Lemma 18. Let be disjoint simple connected graphs, where and . Let be the graph obtained from and by adding new edges . Then

From Lemmas 16, 17, and 18, we have the following theorem.

Theorem 19. The characteristic polynomial of is where , , and is a path of length .

As for the Laplacian spectral radius of , we only need to consider the greatest root of

In general, the Laplacian spectral radius is difficult to calculate although the characteristic polynomial can be identified by Chebvshev polynomials. Thus, we only give some results in some cases.

Lemma 20. For a univariate cubic equation , we can obtain the roots as follows: where , and .

Proof. The lemma is easy to prove.

Theorem 21. If , then Moreover, if , then .

Proof. In this case, for . By (20), we have Hence, .
Let , and then we have for and , and thus has a root in , for and , and thus has a root in , for and , and thus has a root in . Therefore, the equation has three different real roots.
Let , and then the equation can translate to , where and .
Let , where . Then Thus, based on the Lemma 20, we can obtain three roots: For , then and . Thus, , . So the largest root of the univariate cubic equation is For , we have If , then it is easy to prove that . Thus, .

From Theorems 14, 15, and 21, we have the following corollaries.

Corollary 22. Let be a tree for vertices. Then and equality holds if and only if , the star with vertices.

Corollary 23. Let be a tree for vertices with and . Then and equality holds if and only if .

Corollary 24. Let be a tree for vertices. Let denote one maximum matching of and suppose that . Then and equality holds if and only if .

Proof. Let denote the cardinality of the pendant vertices of . For , we have . By Theorems 14 and 15, we have , and equality holds if and only if .
is shown in Figure 7. Since , by Theorem 21 we have

524162.fig.007
Figure 7

Corollary 25. Let be any tree for vertices and . Then and equality holds if and only if .

Acknowledgments

The authors would like to thank the anonymous referees for valuable suggestions and corrections, which have improved the paper. This research was supported by the Scientific Research Foundation of Huaqiao University (10HZR26) and the Natural Science Foundation of Fujian Province (Z0511028).

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