Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

Abstract

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 𝐺=(𝑉,𝐸) be a simple connected graph on 𝑛 vertices and let 𝜇(𝐺) be the largest Laplacian eigenvalue (i.e., the spectral radius) of 𝐺. In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius of 𝐺.

1. Introduction

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 [1], 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 [2]) 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 [3]).

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.

2. Preliminaries

Let 𝐺=(𝑉,𝐸) be a simple undirected graph on 𝑛 vertices. The Laplacian matrix of 𝐺 is the 𝑛×𝑛 matrix 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺), where 𝐴(𝐺) is the adjacency and 𝐷(𝐺) is the diagonal matrix of vertex degrees. It is well known that 𝐿(𝐺) is a positive semidefinite matrix and that (0,𝐞) is an eigenpair of 𝐿(𝐺) where 𝐞 is the all-ones vector. In [4], some of the many results known for Laplacian matrices are given. The spectrum of 𝐿(𝐺) is {𝜇₁(𝐺),𝜇₂(𝐺),…,𝜇_𝑛(𝐺)}, where 𝜇₁(𝐺)≥𝜇₂(𝐺)≥⋯≥𝜇_𝑛(𝐺)=0. The largest eigenvalue 𝜇₁(𝐺) is called the Laplacian spectral radius of the graph 𝐺, denoted by 𝜇(𝐺). For a star graph of order 𝑛, the spectrum is {𝑛,1,1,…,1_𝑛−2,0}.

We recall that upper bounds of the spectral radius of 𝐿(𝐺). It is a well-known fact that 𝜇(𝐺)≤2Δ with equality if and only if 𝐺 is bipartite regular. Shi [5] gave an upper bound for the Laplacian spectral radius of irregular graphs as follows.

Let 𝐺 be a connected irregular graph of order 𝑛 with maximum degree Δ and diameter 𝐷. Then 𝜇(𝐺)<2Δ−2/(2𝐷+1)𝑛.

Li et al. [6] improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs. They show the following result: 𝜇(𝐺)1<2Δ−.𝑛𝐷 (1)

Dyilek Maden and Buyukkose [7] proved the following.

Let 𝐺 be a simple graph. Then, 𝑠𝑚+√𝑛−1≤𝜇₁(𝐺)≤𝑚+𝑠√𝑛−1, (2) where 𝑚=(∑^𝑛_𝑖=1𝑑_𝑖(𝑑_𝑖+1))/𝑛, and 𝑠²=∑^𝑛_𝑖=1𝑑²_𝑖+𝑑_𝑖²𝑛+2∑_{𝑖<𝑗,𝑖∼𝑗}𝑑_𝑖+𝑑_𝑗𝑑_𝑖+𝑑_𝑗−2𝑁_𝑖∩𝑁_𝑗𝑛+2∑_𝑖<𝑗𝑁_𝑖∩𝑁_𝑗²𝑛−𝑚². (3)

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 𝜇(𝐺).

3. Some Useful Lemmas

In the proof of several theorems we will use the following lemmas.

Lemma 1 (see [8]). Let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges; then ^𝑛_𝑖=1𝑑²_𝑖≤2𝑚(2𝑚+(𝑛−1)(Δ−𝛿)).𝑛+Δ−𝛿 (4) Equality holds if and only if 𝐺=𝐵_𝑛,𝑡 for some 1≤𝑡≤𝑛 or 𝐺 is regular, where 𝐵_𝑛,𝑡 denotes the graph on 𝑛 vertices with exactly 𝑡 vertices of degree 𝑛−1 and the remaining of 𝑛−𝑡 vertices forming an independent set. Notice that 𝐵_𝑛,1=𝐾_1,𝑛−1 and 𝐵_𝑛,𝑛=𝐾_𝑛.

Let 𝑀 be an 𝑚×𝑛 matrix. Then 𝑠_𝑖(𝑀) will denote the 𝑖th row sum of 𝑀; that is, 𝑠_𝑖(𝑀)=∑^𝑛_𝑗=1𝑀_𝑖𝑗, where 1≤𝑖≤𝑚.

Lemma 2 (see [9]). Let 𝐺 be a connected 𝑛-vertex graph and 𝐴 its adjacency matrix, with spectral radius 𝜌(𝐴). Let 𝑃 be any polynomial. Then min_{(𝐺)𝑖∈𝑉}𝑠_𝑖(𝑃(𝐴))≤𝑃𝜌(𝐴)≤max_{(𝐺)𝑖∈𝑉}𝑠_𝑖(𝑃(𝐴)). (5)

Lemma 3 (see [10]). Let 𝐺 be a simple graph with vertex set 𝑉(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑛}. Let 𝜌 denote the spectral radius of the line graph 𝑙(𝐺) of 𝐺. Then the inequality 𝜇(𝐺)≤2+𝜌 (6) holds, and the equality occurs if and only if 𝐺 is a bipartite graph.

4. Main Results

In this section, we consider simple connected graph with 𝑛 vertices. The main result of the paper is the following theorem.

Theorem 4. Let 𝐺 be a graph with 𝑛 vertices and 𝑚 edges; then 𝜇(𝐺)≤1()𝑚(𝑛())×2()𝑛−12𝑚+𝑛−2𝑛−1−2𝑚Δ−𝛿+1𝑛+Δ−𝛿^1/2, (7) with equality if and only if 𝐺 is the star 𝐾_1,𝑛−1 or the complete graph 𝐾_𝑛.

Proof. Since ^𝑛_𝑖=1𝜇_𝑖(𝐺)=tr(𝐿(𝐺))=2𝑚, (8) where tr(𝐿(𝐺)) denotes the trace of 𝐿(𝐺). Notice that 𝜇²₁(𝐺),𝜇²₂(𝐺),…,𝜇²_𝑛(𝐺) are eigenvalues of 𝐿²(𝐺); hence we have ^𝑛_𝑖=1𝜇²_𝑖(𝐺)=tr𝐿²(𝐺)=2𝑚+^𝑛_𝑖=1𝑑²_𝑖. (9) According to the Cauchy-Schwarz inequality, we have (𝑛−2)𝜇²₂(𝐺)+𝜇²₃(𝐺)+⋯+𝜇²_𝑛−1(𝐺)≥𝜇₂(𝐺)+𝜇₃(𝐺)+⋯+𝜇_𝑛−1(𝐺)². (10) That is, (𝑛−2)2𝑚−𝜇²(𝐺)+^𝑛_𝑖=1𝑑²_𝑖≥2𝑚−𝜇(𝐺)². (11) By means of Lemma 1, we obtain 2𝑚−𝜇(𝐺)²≤(𝑛−2)2𝑚−𝜇(𝐺)²+()⋅((𝑛−22𝑚2𝑚+𝑛−1)(Δ−𝛿)).𝑛+Δ−𝛿 (12) Suppose that 𝑡=(𝑛−2)⋅(2𝑚(2𝑚+(𝑛−1)(Δ−𝛿)))/(𝑛+Δ−𝛿); simplifying the inequality above, we get (𝑛−1)𝜇(𝐺)²−4𝑚𝜇(𝐺)+4𝑚²−2𝑚(𝑛−2)−𝑡≤0. (13) Hence, we have 2𝑚−𝑤(𝐺)≤𝑛−1≤𝜇2𝑚+𝑤,𝑛−1 (14) where 𝑤=(𝑛−1)𝑡+2𝑚(𝑛−1)(𝑛−2)−4𝑚²(𝑛−2).
Let 𝐺=𝐾_1,𝑛−1 and then 𝑚=𝑛−1, 𝑡=𝑛(𝑛−1)(𝑛−2); 𝑤=(𝑛−1)(𝑛−2); we have 2𝑚+𝑤=2()+()𝑛−1𝑛−1𝑛−1)(𝑛−2(𝐺).𝑛−1=𝑛=𝜇 (15)
Next we examine the complete graph 𝐾_𝑛. The complete graph has 𝑛 vertices and 𝑛(𝑛−1)/2 edges, Δ=𝑛−1=𝛿.
Thus, 𝑡=𝑛(𝑛−1)²(𝑛−2); hence, 𝑤=0; we get 2𝑚+𝑤(𝐺).𝑛−1=𝑛=𝜇 (16)
Equality holds on the right in (14) if and only if 𝐺 is the star 𝐾_1,𝑛−1 or the complete graph 𝐾_𝑛.
This completes the proof of the theorem.

Example 5. In this example we illustrate the technique of Theorem 4. Consider the graph 𝐺 on 6 vertices and 8 edges in Figure 1; this graph has the largest degree Δ=3 and the smallest degree 𝛿=2.

Figure 1 

The graph 𝐺 on 6 vertices.

Now we estimate the largest eigenvalue 𝜇(𝐺) with Theorem 4. Applying this upper bound on 𝜇(𝐺), it follows that 0≤𝜇(𝐺)≤6.4. (17) By a straightforward calculation, we show that the Laplacian eigenvalues of 𝐺 are 𝜇₁(𝐺)=3+√3,𝜇₂(𝐺)=4,𝜇₃(𝐺)𝜇=4,₄(𝐺)=2,𝜇₅(𝐺)=3−√3,𝜇₆(𝐺)=0. (18) Clearly, 𝜇₁(𝐺)=3+√3<6.4 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 𝐺, that is, associated with the largest and the second largest degree Δ^′, the smallest degree of 𝐺, respectively.

Theorem 6. Let 𝐺=(𝑉,𝐸) be a simple graph with 𝑛 vertices and 𝑚 edges; then 𝜇(𝐺)≤Δ^′+Δ+𝛿−1_′Δ+𝛿−1+4²−Δ⋅Δ^′+4𝑚−2(𝑛−1)𝛿^1/2×(2)⁻¹ (19) with equality if and only if 𝐺 is a regular bipartite graph.

Proof. Let 𝐴=(𝑎_𝑖𝑗) be the adjacency matrix of a graph 𝐺 with vertices 1,2,…,𝑛; let 𝐴^𝑘=(𝑎^(𝑘)_𝑖𝑗); further, let 𝑁_𝑖(𝐴^𝑘) denote the number of walks of length 𝑘 starting at vertex 𝑖. Hence 𝑁_𝑖(𝐴²)=∑_𝑗∼𝑖𝑑_𝑗.
Similarly, for the adjacency matrix 𝐵 of line graph 𝑙(𝐺) of 𝐺, let 𝑁^′_𝑢(𝐵^𝑘) denote the number of walks of length 𝑘 starting at vertex 𝑢 and let 𝑑_𝑙(𝐺)(𝑖𝑗) denote the degree of the vertex 𝑖𝑗 in line graph 𝑙(𝐺); then we have 𝑁^′_𝑖𝑗(𝐵)=𝑑_𝑙(𝐺)𝑖𝑗=𝑑_𝑖+𝑑_𝑗−2. (20) It can easily be seen that 𝑁^′_𝑖𝑗𝐵²=_{𝑝𝑞∼𝑖𝑗}𝑑_𝑙(𝐺)𝑝𝑞=_{𝑝𝑞∼𝑖𝑗}𝑑_𝑝+𝑑_𝑞−2; (21) that is, _𝑞∼𝑖𝑑_𝑞+𝑑_𝑖−2+_𝑝∼𝑗𝑑_𝑝+𝑑_𝑗−2−2𝑑_𝑖+𝑑_𝑗−2=𝑑²_𝑖−2𝑑_𝑖+_𝑞∼𝑖𝑑_𝑞+𝑑²_𝑗−2𝑑_𝑗+_𝑝∼𝑗𝑑_𝑝−2𝑑_𝑖+𝑑_𝑗−2≤Δ𝑑_𝑖+Δ^′𝑑_𝑗+4𝑚−3𝑑_𝑖+𝑑_𝑗−_{𝑞̸∼𝑖}𝑑_𝑞−_{𝑝̸∼𝑗}𝑑_𝑝−2𝑑_𝑖+𝑑_𝑗−2≤Δ^′𝑑_𝑖+𝑑_𝑗+Δ−Δ^′Δ+4𝑚−3𝑑_𝑖+𝑑_𝑗−𝑛−𝑑_𝑖−1𝛿−𝑛−𝑑_𝑗−1𝛿−2𝑑_𝑖+𝑑_𝑗=Δ−2_′+𝛿−5𝑑_𝑖+𝑑_𝑗−2+Δ²+(2−Δ)Δ^′−2(𝑛−2)𝛿+4𝑚−6. (22) According to (20), we have 𝑁^′_𝑖𝑗𝐵²−Δ^′+𝛿−5𝐵≤Δ²+(2−Δ)Δ^′−2(𝑛−2)𝛿+4𝑚−6. (23) Using Lemma 2, we obtain 𝜌²−Δ^′+𝛿−5𝜌≤Δ²+(2−Δ)Δ^′−2(𝑛−2)𝛿+4𝑚−6. (24) From the inequality above, we have 𝜌≤Δ^′+Δ+𝛿−5_′+𝛿−5²+4Δ²+(2−Δ)Δ^′−2(𝑛−2)𝛿+4𝑚−6^1/2×(2)⁻¹=Δ^′+Δ+𝛿−5_′+𝛿−1²+4Δ²−Δ⋅Δ^′−2(𝑛−1)𝛿+4𝑚^1/2×(2)⁻¹. (25) Using Lemma 3, we have 𝜇(𝐺)≤Δ^′+Δ+𝛿−1_′Δ+𝛿−1+4²−Δ⋅Δ^′+4𝑚−2(𝑛−1)𝛿^1/2×(2)⁻¹. (26)
If equality holds in (19), applying Lemma 3, hence we deduce that 𝐺 is a bipartite graph. And by Lemma 2, for any 𝑖𝑗∈𝐸(𝐺), the equality in (25) occurs. That is, for any 𝑖𝑗, the equality in (22) holds. Then, we have Δ=Δ^′=𝛿; (27) that is, 𝐺 is a regular graph; hence 𝐺 is a regular bipartite.
More generally, if 𝐺 is a regular bipartite graph, then by verifying straightforward (19), the equality holds.

Acknowledgments

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.