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 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.


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 secondsmallest 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.

Preliminaries
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 (0, e) is an eigenpair of () where e is the all-ones vector.In [4], some of the many results known for Laplacian matrices are given.The spectrum of () is { 1 (),  2 (), . . .,   ()}, where  1 () ≥  2 () ≥ ⋅ ⋅ ⋅ ≥   () = 0.The largest eigenvalue  1 () is called the Laplacian spectral radius of the graph , denoted by ().For a star graph of order , the spectrum is {, 1, 1, . . ., 1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ We recall that upper bounds of the spectral radius of ().It is a well-known fact that () ≤ 2Δ with equality if and only if  is bipartite regular.Shi [5] gave an upper bound for the Laplacian spectral radius of irregular graphs as follows.
Li et al. [6] improve Shi's upper bound for the Laplacian spectral radius of irregular graphs.They show the following result: Dyilek Maden and Buyukkose [7] proved the following.
Let  be a simple graph.Then, where  = (∑  =1   (  + 1))/, 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 ().

Some Useful Lemmas
In the proof of several theorems we will use the following lemmas.
Lemma 1 (see [8]).Let  be a connected graph with  vertices and  edges; then Equality holds if and only if  =  , for some 1 ≤  ≤  or  is regular, where  , denotes the graph on  vertices with exactly  vertices of degree −1 and the remaining of − vertices forming an independent set.Notice that  ,1 =  1,−1 and  , =   .

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  1,−1 or the complete graph   . Proof.
This completes the proof of the theorem.
Example 5.In this example we illustrate the technique of Theorem 4. Consider the graph  on 6 vertices and 8 edges in Figure 1; this graph has the largest degree Δ = 3 and the smallest degree  = 2.
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 Δ  , 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.
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 (24)