# Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

Authors: Houqing Zhou; Youzhuan Xu
Journal: Mathematical Problems in Engineering
 (2013)

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## 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. LetG=(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.


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## Body

## 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

LetG=(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),…,μn(G)}, where μ1(G)≥μ2(G)≥⋯≥μn(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 ofL(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.LetG 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:
(1)μ(G)<2Δ-1nD.Dyilek Maden and Buyukkose [7] proved the following.LetG be a simple graph. Then,
(2)m+sn-1≤μ1(G)≤m+sn-1,
where m=(∑i=1n‍di(di+1))/n, and
(3)s2=∑i=1n(di2+di)2n+2∑i<j,i~j(di+dj)(di+dj-2|Ni∩Nj|)n+2∑i<j|Ni∩Nj|2n-m2.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. Section3 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]).
LetG be a connected graph with n vertices and m edges; then
(4)∑i=1n‍di2≤2m(2m+(n-1)(Δ-δ))n+Δ-δ.
Equality holds if and only if G=Bn,t for some 1≤t≤n or G is regular, where Bn,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 Bn,1=K1,n-1 and Bn,n=Kn.LetM be an m×n matrix. Then si(M) will denote the ith row sum of M; that is, si(M)=∑j=1n‍Mij, where 1≤i≤m.Lemma 2 (see [9]).
LetG be a connected n-vertex graph and A its adjacency matrix, with spectral radius ρ(A). Let P be any polynomial. Then
(5)mini∈V(G)si(P(A))≤P(ρ(A))≤maxi∈V(G)si(P(A)).Lemma 3 (see [10]).
LetG be a simple graph with vertex set V(G)={v1,v2,…,vn}. Let ρ denote the spectral radius of the line graph l(G) of G. Then the inequality
(6)μ(G)≤2+ρ
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 withn vertices. The main result of the paper is the following theorem.Theorem 4.
LetG be a graph with n vertices and m edges; then
(7)μ(G)≤1n-1(2(Δ-δ+1)n+Δ-δ)1/22m+(2(Δ-δ+1)n+Δ-δ(n-2)m(n(n-1)-2m)hhhhhhhh×2(Δ-δ+1)n+Δ-δ)1/2),
with equality if and only if G is the star K1,n-1 or the complete graph Kn.Proof.
Since(8)∑i=1n‍μi(G)=tr(L(G))=2m,
where tr(L(G)) denotes the trace of L(G). Notice that μ12(G),μ22(G),…,μn2(G) are eigenvalues of L2(G); hence we have
(9)∑i=1n‍μi2(G)=tr(L2(G))=2m+∑i=1n‍di2.
According to the Cauchy-Schwarz inequality, we have
(10)(n-2)(μ22(G)+μ32(G)+⋯+μn-12(G))≥(μ2(G)+μ3(G)+⋯+μn-1(G))2.
That is,
(11)(n-2)(2m-μ2(G)+∑i=1n‍di2)≥(2m-μ(G))2.
By means of Lemma 1, we obtain
(12)(2m-μ(G))2≤(n-2)(2m-μ(G)2)+(n-2)·2m(2m+(n-1)(Δ-δ))n+Δ-δ.
Suppose that t=(n-2)·(2m(2m+(n-1)(Δ-δ)))/(n+Δ-δ); simplifying the inequality above, we get
(13)(n-1)μ(G)2-4mμ(G)+4m2-2m(n-2)-t≤0.
Hence, we have
(14)2m-wn-1≤μ(G)≤2m+wn-1,
where w=(n-1)t+2m(n-1)(n-2)-4m2(n-2).
LetG=K1,n-1 and then m=n-1, t=n(n-1)(n-2); w=(n-1)(n-2); we have
(15)2m+wn-1=2(n-1)+(n-1)(n-2)n-1=n=μ(G).
Next we examine the complete graphKn. 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
(16)2m+wn-1=n=μ(G).
Equality holds on the right in (14) if and only if G is the star K1,n-1 or the complete graph Kn.
This completes the proof of the theorem.Example 5.
In this example we illustrate the technique of Theorem4. 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 graphG on 6 vertices.Now we estimate the largest eigenvalueμ(G) with Theorem 4. Applying this upper bound on μ(G), it follows that
(17)0≤μ(G)≤6.4.
By a straightforward calculation, we show that the Laplacian eigenvalues of G are
(18)μ1(G)=3+3,μ2(G)=4,μ3(G)=4,μ4(G)=2,μ5(G)=3-3,μ6(G)=0.
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 Theorem6 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.
LetG=(V,E) be a simple graph with n vertices and m edges; then
(19)μ(G)≤(+4(Δ2-Δ·Δ′+4m-2(n-1)δ))1/2Δ′+δ-1+((Δ′+δ-1)+4(Δ2-Δ·Δ′+4m-2(n-1)δ))1/2)×(2)-1
with equality if and only if G is a regular bipartite graph.Proof.
LetA=(aij) be the adjacency matrix of a graph G with vertices 1,2,…,n; let Ak=(aij(k)); further, let Ni(Ak) denote the number of walks of length k starting at vertex i. Hence Ni(A2)=∑j~i‍dj.
Similarly, for the adjacency matrixB of line graph l(G) of G, let Nu′(Bk) denote the number of walks of length k starting at vertex u and let dl(G)(ij) denote the degree of the vertex ij in line graph l(G); then we have
(20)Nij′(B)=dl(G)(ij)=di+dj-2.
It can easily be seen that
(21)Nij′(B2)=∑pq~ij‍dl(G)(pq)=∑pq~ij‍(dp+dq-2);
that is,
(22)∑q~i‍(dq+di-2)+∑p~j‍(dp+dj-2)-2(di+dj-2)=di2-2di+∑q~i‍dq+dj2-2dj+∑p~j‍dp-2(di+dj-2)≤Δdi+Δ′dj+4m-3(di+dj)-∑q≁i‍dq-∑p≁j‍dp-2(di+dj-2)≤Δ′(di+dj)+(Δ-Δ′)Δ+4m-3(di+dj)-(n-di-1)δ-(n-dj-1)δ-2(di+dj-2)=(Δ′+δ-5)(di+dj-2)+Δ2+(2-Δ)Δ′-2(n-2)δ+4m-6.
According to (20), we have
(23)Nij′((B2-(Δ′+δ-5)B))≤Δ2+(2-Δ)Δ′-2(n-2)δ+4m-6.
Using Lemma 2, we obtain
(24)ρ2-(Δ′+δ-5)ρ≤Δ2+(2-Δ)Δ′-2(n-2)δ+4m-6.
From the inequality above, we have
(25)ρ≤(+4(Δ2+(2-Δ)Δ′-2(n-2)δ+4m-6))1/2Δ′+δ-5+((Δ′+δ-5)2+4(Δ2+(2-Δ)Δ′-2(n-2)δ+4m-6))1/2)×(2)-1=(+4(Δ2-Δ·Δ′-2(n-1)δ+4m))1/2Δ′+δ-5+((Δ′+δ-1)2+4(Δ2-Δ·Δ′-2(n-1)δ+4m))1/2)×(2)-1.
Using Lemma 3, we have
(26)μ(G)≤(+4(Δ2-Δ·Δ′+4m-2(n-1)δ))1/2Δ′+δ-1+((Δ′+δ-1)+4(Δ2-Δ·Δ′+4m-2(n-1)δ))1/2)×(2)-1.
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
(27)Δ=Δ′=δ;
that is, G is a regular graph; hence G is a regular bipartite.
More generally, ifG is a regular bipartite graph, then by verifying straightforward (19), the equality holds.



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