Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs
The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. Let be a simple connected graph on vertices and let be the largest Laplacian eigenvalue (i.e., the spectral radius) of . In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius of .
The eigenvalue spectrum of the Laplacian matrix of a network provides valuable information regarding the behavior of many dynamical processes taking place on the network. In , Pecora and Carroll related the problem of synchronization in a network of coupled oscillators to the largest and second-smallest Laplacian eigenvalues (usually denoted by Laplacian spectral radius and spectral gap, resp.) of the network. More recently, Dorfler and Bullo (see ) derived conditions for transient stability in power networks in terms of the spectral gap of the Laplacian matrix. Apart from their applicability to the problems of synchronization and transient stability analysis, the Laplacian eigenvalues are also relevant in the analysis of many distributed estimation and control problems (see ).
Understanding the relationship between the structure of a complex network and the behavior of dynamical processes taking place in it is a central question in the research field of network science. Since the behavior of many networked dynamical processes is closely related to the Laplacian eigenvalues, it is of interest to study the relationship between structural features of the network and its Laplacian eigenvalues. In this paper, we mainly study the spectral radius of the Laplacian matrix.
Let be a simple undirected graph on vertices. The Laplacian matrix of is the matrix , where is the adjacency and is the diagonal matrix of vertex degrees. It is well known that is a positive semidefinite matrix and that is an eigenpair of where is the all-ones vector. In , some of the many results known for Laplacian matrices are given. The spectrum of is , where . The largest eigenvalue is called the Laplacian spectral radius of the graph , denoted by . For a star graph of order , the spectrum is .
We recall that upper bounds of the spectral radius of . It is a well-known fact that with equality if and only if is bipartite regular. Shi  gave an upper bound for the Laplacian spectral radius of irregular graphs as follows.
Let be a connected irregular graph of order with maximum degree and diameter . Then .
Li et al.  improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs. They show the following result:
Dyilek Maden and Buyukkose  proved the following.
Let be a simple graph. Then, where , and
In this paper, we continue to consider the upper bounds for the Laplacian spectral radius of graphs. The rest of the paper is organized as follows. Section 3 contains some lemmas which play a fundamental role. Section 4 contains two theorems on the upper bounds of .
3. Some Useful Lemmas
In the proof of several theorems we will use the following lemmas.
Lemma 1 (see ). Let be a connected graph with vertices and edges; then Equality holds if and only if for some or is regular, where denotes the graph on vertices with exactly vertices of degree and the remaining of vertices forming an independent set. Notice that and .
Let be an matrix. Then will denote the th row sum of ; that is, , where .
Lemma 2 (see ). Let be a connected -vertex graph and its adjacency matrix, with spectral radius . Let be any polynomial. Then
Lemma 3 (see ). Let be a simple graph with vertex set . Let denote the spectral radius of the line graph of . Then the inequality holds, and the equality occurs if and only if is a bipartite graph.
4. Main Results
In this section, we consider simple connected graph with vertices. The main result of the paper is the following theorem.
Theorem 4. Let be a graph with vertices and edges; then with equality if and only if is the star or the complete graph .
where denotes the trace of . Notice that are eigenvalues of ; hence we have
According to the Cauchy-Schwarz inequality, we have
By means of Lemma 1, we obtain
Suppose that ; simplifying the inequality above, we get
Hence, we have
Let and then , ; ; we have
Next we examine the complete graph . The complete graph has vertices and edges, .
Thus, ; hence, ; we get
Equality holds on the right in (14) if and only if is the star or the complete graph .
This completes the proof of the theorem.
Now we estimate the largest eigenvalue with Theorem 4. Applying this upper bound on , it follows that By a straightforward calculation, we show that the Laplacian eigenvalues of are Clearly, holds.
It is easy to see that we can use the method to estimate the upper bound of the largest Laplacian eigenvalue.
The following Theorem 6 is associated with edge and degree of graph , that is, associated with the largest and the second largest degree , the smallest degree of , respectively.
Theorem 6. Let be a simple graph with vertices and edges; then with equality if and only if is a regular bipartite graph.
Proof. Let be the adjacency matrix of a graph with vertices ; let ; further, let denote the number of walks of length starting at vertex . Hence .
Similarly, for the adjacency matrix of line graph of , let denote the number of walks of length starting at vertex and let denote the degree of the vertex in line graph ; then we have It can easily be seen that that is, According to (20), we have Using Lemma 2, we obtain From the inequality above, we have Using Lemma 3, we have
If equality holds in (19), applying Lemma 3, hence we deduce that is a bipartite graph. And by Lemma 2, for any , the equality in (25) occurs. That is, for any , the equality in (22) holds. Then, we have that is, is a regular graph; hence is a regular bipartite.
More generally, if is a regular bipartite graph, then by verifying straightforward (19), the equality holds.
The authors thank Professor Miguel A. F. Sanjuán for his useful comments and suggestions. Project is supported by Hunan Provincial Natural Science Foundation of China no. 13JJ3118 and by Scientific Research Fund of Shaoyang Science & Technology no. M230.
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