Abstract

Let be weighted graphs, as the graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of Laplacian matrix of a graph . We obtain two upper bounds for the largest Laplacian eigenvalue of weighted graphs and we compare these bounds with previously known bounds.

1. Introduction

Let be simple graphs, as graphs which have no loops or parallel edges such that is a finite set of vertices and is a set of edges.

A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weight matrices are assumed to be of same order and to be positive matrix. In this paper, by “weighted graph” we mean “a weighted graph with each of its edges bearing a positive definite matrix as weight,” unless otherwise stated.

The notations to be used in paper are given in the following.

Let be a weighted graph on vertices. Denote by the positive definite weight matrix of order of the edge , and assume that . We write if vertices and are adjacent. Let . be the weight matrix of the vertex .

The Laplacian matrix of a graph is defined as , where

The zero denotes the zero matrix. Hence is square matrix of order . Let denote the largest eigenvalue of . In this paper we also use to avoid the confusion that is the spectral radius of matrix. If is the disjoint union of two nonempty sets and such that every vertex in has the same and every vertex in has the same , then is called a weight-semiregular graph. If in weight semiregular graph, then is called a weighted-regular graph.

Upper and lower bounds for the largest Laplacian eigenvalue for unweighted graphs have been investigated to a great extent in the literature. Also there are some studies about the bounds for the largest Laplacian eigenvalue of weighted graphs [1–3]. The main result of this paper, contained in Section 2, gives two upper bounds on the largest Laplacian for weighted graphs, where the edge weights are positive definite matrices. These upper bounds are attained by the same methods in [1–3]. We also compare the upper bounds with the known upper bounds in [1–3]. We also characterize graphs which achieve the upper bound. The results clearly generalize some known results for weighted and unweighted graphs.

2. The Known Upper Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

In this section, we present the upper bounds for the largest Laplacian eigenvalue of weighted graphs and very useful lemmas to prove theorems.

Theorem 1 (Horn and Johnson [4]). Let be Hermitian, and let the eigenvalues of be ordered such that . Then, for all .

Lemma 2 (Horn and Johnson [4]). Let be a Hermitian matrix with eigenvalues ; then for any , , Equality holds if and only if is an eigenvector of corresponding to and for some .

Lemma 3 (see [1]). Let be a -semiregular bipartite graph of order such that the first vertices of the same largest eigenvalue and the remaining vertices of the same largest eigenvalue . Also let be a common eigenvector of corresponding to the largest eigenvalue for all , , where for all . Then is the largest eigenvalue of and the corresponding eigenvector is

Theorem 4 (see [1]). Let be a simple connected weighted graph. Then where is the positive definite weight matrix of order of the edge . Moreover equality holds in (6) if and only if(i)is a weight-semiregular bipartite graph,(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Theorem 5 (see [2]). Let be a simple connected weighted graph. Then where is the positive definite weight matrix of order of the edge . Moreover equality holds in (7) if and only if(i) is a bipartite semiregular graph;(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Corollary 6 (see [2]). Let be a simple connected weighted graph where each edge weight is a positive number. Then where and is the weight of vertex . Moreover equality holds if and only if is a bipartite regular graph.

Corollary 7 (see [2]). Let be a simple connected weighted graph where each edge weight is a positive number. Then where and is the weight of vertex . Moreover equality holds if and only if is a bipartite semiregular graph.

Theorem 8 (see [2]). Let be a simple connected weighted graph. Then where and is the positive definite weight matrix of order of the edge . Moreover equality holds in (10) if and only if(i) is a weighted-regular graph or is a weight-semiregular bipartite graph;(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Corollary 9 (see [2]). Let be a simple connected weighted graph where each edge weight is a positive number. Then where and is the weight of vertex . Moreover equality holds if and only if is a bipartite semiregular graph or is a bipartite regular graph.

Theorem 10 (see [3]). Let be a simple connected weighted graph. Then where is the positive definite weight matrix of order of the edge and is the set of common neighbours of and . Moreover equality holds in (12) if and only if(i) is a weight-semiregular bipartite graph;(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Corollary 11 (see [3]). Let be a simple connected weighted graph where each edge weight is a positive number. Then Moreover equality holds if and only if is a bipartite semiregular graph.

3. Two Upper Bounds on the Largest Laplacian Eigenvalue of Weighted Graphs

In this section we present two upper bounds for the largest eigenvalue of weighted graphs and compare the bounds with some examples.

Theorem 12. Let be a simple connected weighted graph. Thenwhere is the positive definite weight matrix of order of the edge . Moreover equality holds in (14) if and only if(i) is a weighted-regular graph or is a weight-semiregular bipartite graph;(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Proof. Let be an eigenvector corresponding to the largest eigenvalue of . We assume that is the vector component of such that Since is nonzero, so is . Let be. The th element of is
We have
From the th equation of (18), we have that is,
From (23) we have that is,
From the th equation of (18), we get that is,
Similarly, from (30) we get that is,
So, from (25) and (32) we have
Hence we get that is, that is,
This completes the proof of (14).
Now suppose that equality holds in (14). Then all inequalities in the previous argument must be equalities.
From equality in (23), we get
Since , we get that for all , . From equality in (22) and Lemma 2, we get that is an eigenvector of for the largest eigenvalue . Hence we say that for some , for any , .
On the other hand, from (37) we get that is,
From equality in (21), we have
Since , from (40) we get Hence we get from equalities in (41). Therefore we have
Similarly from equality in (29), we get that is an eigenvector of for the largest eigenvalue . Hence we say that for some , for any , . From equality in (16) we have that is, that is,
Applying the same methods as previously, we get
Therefore we have
For
Hence we take that and from (43), (48), and (49). So, and . Also, since . Further, for any vertex there exists a vertex such that , where is the neighbor of neighbor set of vertex . Therefore and . So . By similar argument we can present that . Continuing the procedure, it is easy to see, since is connected, that and that the subgraphs induced by and , respectively, are empty graphs. Hence is bipartite. Moreover, is a common eigenvector of and for the largest eigenvalue and .
For that is,
Since is an eigenvector of corresponding to the largest eigenvalue of for all , we get that is, that is,
Therefore we get that is constant for all . Similarly we can show that is constant for all .
Hence is a bipartite semiregular graph.
Conversely, suppose that conditions (i)-(ii) of the theorem hold for the graph . Let be -semiregular bipartite graph. Let be a common eigenvector of corresponding to the largest eigenvalue for all . Then we have
By Lemma 3, we get that is,

Corollary 13 (see [1]). Let be a simple connected weighted graph where each edge weight is a positive number. Then Moreover equality holds in (58) if and only if is bipartite semiregular graph.

Proof. We have and for all , . From Theorem 12, we get the required result.

Corollary 14 (see [5]). Let be a simple connected unweighted graph. Then where is the degree of vertex . Moreover equality holds in (59) if and only if is a bipartite regular graph or is a bipartite semiregular graph.

Proof. For unweighted graph, for . Therefore . Using Corollary 6, we get the required results.

Theorem 15. Let be a simple connected weighted graph. Then where is the positive definite weight matrix of order of the edge . Moreover equality holds in (60) if and only if(i) is a weighted-regular bipartite graph;(ii) have a common eigenvector corresponding to the largest eigenvalue for all , .

Proof. Let be an eigenvector corresponding to the largest eigenvalue of . We assume that is the vector component of such that Since is nonzero, so is . Let be. We have
From the th equation of (43), we have that is, Hence we get By the same method, from the th equation of (43), we have that is,
Hence we get
From (49) and (58), we have that is,
This completes the proof of (60).
Now we show the case of equality in (60). By similar method in Theorem 12. In the part of equalit, the necessary condition can show easily. So we will show the sufficient condition.
Suppose that conditions (i)-(ii) of Theorem hold for the graph . We must prove that
Let be regular bipartite graph. Therefore we have for and for such that . Let be a common eigenvector of corresponding to the largest eigenvalue for all . Hence we have
From (71) we get that
On the other hand, the following equation can be easily verified:
Thus is an eigenvalue of . Since is the largest eigenvalue of , we get
So from (74) and (76) we obtain

Corollary 16. Let be a simple connected weighted graph where each edge weight is a positive number. Then Moreover equality holds in (78) if and only if is bipartite semiregular graph.

Proof. We have and for all , . From Theorem 15 we get the required result.

Corollary 17. Let be a simple connected unweighted graph. Then where is the degree of vertex . Moreover equality holds in (79) if and only if is a bipartite regular graph or is a bipartite semiregular graph.

Proof. For unweighted graph, for . Therefore . Using Corollary 16, we get the required results.

Example 18. Let and be a weighted graph where , and each weight is the positive definite matrix of order three. Let , such that each weight is the positive definite matrix of order two. Assume that the following Laplacian matrices of and are as follows:

The largest eigenvalues of and are , rounded two decimal places and the previously mentioned bounds give the following results:

For , we see that the upper bounds in (14) and (60) are better than upper bounds in (6) and (7). But they are not better than upper bounds in (10) and (12) from (81).

For also , we see that upper bounds in (14) and (60) are only better than the upper bound in (6).

Consequently, we cannot exactly compare all the bounds for weighted graphs, where the weights are positive definite matrices. Modifications according to each weight of edges, especially for matrices can be shown.