# Bounds on normalized Laplacian eigenvalues of graphs

## Abstract

Let G be a simple connected graph of order n, where $n\ge 2$. Its normalized Laplacian eigenvalues are $0={\lambda }_{1}\le {\lambda }_{2}\le \cdots \le {\lambda }_{n}\le 2$. In this paper, some new upper and lower bounds on ${\lambda }_{n}$ are obtained, respectively. Moreover, connected graphs with ${\lambda }_{2}=1$ (or ${\lambda }_{n-1}=1$) are also characterized.

MSC:05C50, 15A48.

## 1 Introduction

Let G be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. Its order is $|V\left(G\right)|$, denoted by n, and its size is $|E\left(G\right)|$, denoted by m. In this paper, all graphs are simple connected of order $n\ge 2$. For $v\in V\left(G\right)$, let $d\left(v\right)$ and $N\left(v\right)$ be the degree and the set of neighbors of v, respectively. The maximum and minimum degrees of G are denoted by Δ and δ, respectively.

Let $A\left(G\right)$ and $D\left(G\right)$ be the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The Laplacian and normalized Laplacian matrices of G are defined as $L\left(G\right)=D\left(G\right)-A\left(G\right)$ and $\mathcal{L}\left(G\right)=D{\left(G\right)}^{-1/2}L\left(G\right)D{\left(G\right)}^{-1/2}$, respectively. When only one graph G is under consideration, we sometimes use A, D, L and instead of $A\left(G\right)$, $D\left(G\right)$, $L\left(G\right)$ and $\mathcal{L}\left(G\right)$, respectively. It is easy to see that $\mathcal{L}\left(G\right)$ is a symmetric positive semidefinite matrix and $D{\left(G\right)}^{1/2}\mathbf{1}$ is an eigenvector of $\mathcal{L}\left(G\right)$ with eigenvalue 0, where 1 is the vector with all ones. Thus, the eigenvalues ${\lambda }_{i}\left(G\right)$ ($1\le i\le n$) of $\mathcal{L}\left(G\right)$ (or the normalized Laplacian eigenvalues of G) satisfy

${\lambda }_{n}\left(G\right)\ge \cdots \ge {\lambda }_{2}\left(G\right)\ge {\lambda }_{1}\left(G\right)=0.$

Some of them may be repeated according to their multiplicities. We call ${\lambda }_{k}\left(G\right)$ the k th smallest normalized Laplacian eigenvalue of G. When only one graph G is under consideration, we sometimes write ${\lambda }_{k}$ instead of ${\lambda }_{k}\left(G\right)$, for $1\le k\le n$.

The normalized Laplacian is mentioned briefly in the recent monograph by Cvetković et al. ; however, the standard reference for it is the monograph by Chung , which deals almost entirely with this matrix. The normalized Laplacian eigenvalues can be used to give useful information about a graph . For example, one can obtain the number of connected components from the multiplicity of the eigenvalue 0, the bipartiteness from its ${\lambda }_{n}$ (which is at most 2), as well as the connectivity from its ${\lambda }_{2}$. Moreover, ${\lambda }_{2}$ is also closely related to the discrete Cheeger’s constant, isoperimetric problems, etc. (see ). Chen and Jost  established the relationship between minimum vertex covers and the eigenvalues of the normalized Laplacian on trees. Some upper bounds for ${\lambda }_{n}$ have been introduced by Rojo and Soto  and Banerjee , respectively. For more results on the normalized Laplacian eigenvalues of graphs can be found in [2, 6, 7].

In this paper, some new upper and lower bounds on ${\lambda }_{n}$ of a graph in terms of its maximum degree, covering number etc., are deduced, respectively. Moreover, connected graphs with ${\lambda }_{2}=1$ (or ${\lambda }_{n-1}=1$) are also characterized.

## 2 Preliminaries

Here we recall some basic properties of the eigenvalues and eigenfunctions of the normalized Laplacian matrix of a graph G.

Let $\mathbf{g}:V\left(G\right)\to {\mathbb{R}}^{n}$ which assigns to each vertex v of G a real value $g\left(v\right)$, the coordinate of g according to v. Let $\mathbf{f}={D}^{-1/2}\mathbf{g}$. Then we have

$\frac{{\mathbf{g}}^{T}\mathcal{L}\mathbf{g}}{{\mathbf{g}}^{T}\mathbf{g}}=\frac{{\mathbf{f}}^{T}{D}^{1/2}\mathcal{L}{D}^{1/2}\mathbf{f}}{{\left({D}^{1/2}\mathbf{f}\right)}^{T}{D}^{1/2}\mathbf{f}}=\frac{{\mathbf{f}}^{T}L\mathbf{f}}{{\mathbf{f}}^{T}D\mathbf{f}}=\frac{{\sum }_{uv\in E\left(G\right)}{\left(f\left(u\right)-f\left(v\right)\right)}^{2}}{{\sum }_{v\in V\left(G\right)}d\left(v\right)f{\left(v\right)}^{2}}.$

Thus, the following formula for ${\lambda }_{n}$ is clear:

${\lambda }_{n}=\underset{\mathbf{f}\perp D\mathbf{1}}{sup}\frac{{\sum }_{uv\in E\left(G\right)}{\left(f\left(u\right)-f\left(v\right)\right)}^{2}}{{\sum }_{v\in V\left(G\right)}d\left(v\right)f{\left(v\right)}^{2}}.$
(2.1)

A vector f that satisfies equality in Eq. (2.1) is called a harmonic eigenfunction of associated with ${\lambda }_{n}\left(G\right)$.

Proposition 2.1 ()

Let G be a graph and f be a harmonic eigenfunction of associated with ${\lambda }_{n}\left(G\right)$. Then for any $v\in V\left(G\right)$, we have

$f\left(v\right)-\frac{1}{d\left(v\right)}\sum _{uv\in E\left(G\right)}f\left(u\right)={\lambda }_{n}\left(G\right)f\left(v\right).$

## 3 Main result

We call G a triangulation, if every pair of adjacent vertices of G have at least one common adjacent vertex. A planar graph is called a maximal planar graph if for every pair of nonadjacent vertices u and v of G, the graph $G+uv$ is nonplanar. Lu et al.  and Guo et al.  gave the upper bounds for the Laplacian spectral radius of a triangulation and a maximal planar graph, respectively. For the normalized Laplacian spectral radius, we have the following somewhat similar result.

Theorem 3.1 Let $G=\left(V,E\right)$ be a triangulation of order n. Then

${\lambda }_{n}\le max\left\{\frac{2d\left({v}_{i}\right)-1+\sqrt{4d\left({v}_{i}\right)m\left({v}_{i}\right)-4d\left({v}_{i}\right)+1}}{2d\left({v}_{i}\right)}:{v}_{i}\in V\right\},$

where $m\left({v}_{i}\right)={\sum }_{{v}_{j}\in N\left({v}_{i}\right)}d\left({v}_{j}\right)/d\left({v}_{i}\right)$ is the average 2-degree of the vertex ${v}_{i}$. Moreover, the equality holds if $G\cong {K}_{3}$.

Proof Let $V=\left\{{v}_{1},\dots ,{v}_{n}\right\}$, and let $\mathbf{f}={\left(f\left({v}_{1}\right),\dots ,f\left({v}_{n}\right)\right)}^{T}$ be the harmonic eigenfunction of $\mathcal{L}\left(G\right)$ corresponding to ${\lambda }_{n}$. Then by Proposition 2.1, we have for each ${v}_{i}\in V$,

$d\left({v}_{i}\right)\left(1-{\lambda }_{n}\right)f\left({v}_{i}\right)=\sum _{{v}_{i}{v}_{j}\in E}f\left({v}_{j}\right).$

Hence by the Lagrange identity, we have for each ${v}_{i}$,

$d{\left({v}_{i}\right)}^{2}{\left(1-{\lambda }_{n}\right)}^{2}f{\left({v}_{i}\right)}^{2}=d\left({v}_{i}\right)\sum _{{v}_{i}{v}_{j}\in E}f{\left({v}_{j}\right)}^{2}-\underset{{v}_{j},{v}_{k}\in N\left({v}_{i}\right)}{\sum _{1\le j

Sum over ${v}_{i}$ to obtain

$\begin{array}{c}\sum _{i=1}^{n}d{\left({v}_{i}\right)}^{2}{\left(1-{\lambda }_{n}\right)}^{2}f{\left({v}_{i}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{i=1}^{n}d\left({v}_{i}\right)\sum _{{v}_{i}{v}_{j}\in E}f{\left({v}_{j}\right)}^{2}-\sum _{i=1}^{n}\underset{{v}_{j},{v}_{k}\in N\left({v}_{i}\right)}{\sum _{1\le j
(3.1)

where $m\left({v}_{i}\right)={\sum }_{{v}_{j}\in N\left({v}_{i}\right)}d\left({v}_{j}\right)/d\left({v}_{i}\right)$.

Note that G is a triangulation. Then by Eq. (2.1), we have

$\sum _{i=1}^{n}\underset{{v}_{j},{v}_{k}\in N\left({v}_{i}\right)}{\sum _{1\le j
(3.2)

Thus, combining Eqs. (3.1) and (3.2), we have

$\sum _{i=1}^{n}\left[d{\left({v}_{i}\right)}^{2}{\left(1-{\lambda }_{n}\right)}^{2}-d\left({v}_{i}\right)m\left({v}_{i}\right)+{\lambda }_{n}d\left({v}_{i}\right)\right]f{\left({v}_{i}\right)}^{2}\le 0.$

This implies that there exists at least one vertex ${v}_{i}$ such that

$d{\left({v}_{i}\right)}^{2}{\left(1-{\lambda }_{n}\right)}^{2}-d\left({v}_{i}\right)m\left({v}_{i}\right)+{\lambda }_{n}d\left({v}_{i}\right)\le 0.$

That is,

${\lambda }_{n}\le max\left\{\frac{2d\left({v}_{i}\right)-1+\sqrt{4d\left({v}_{i}\right)m\left({v}_{i}\right)-4d\left({v}_{i}\right)+1}}{2d\left({v}_{i}\right)}:{v}_{i}\in V\right\}.$

For $G={K}_{3}$, it is easy to check that the equality holds. □

Furthermore, we have the following more general result.

Theorem 3.2 Let $G=\left(V,E\right)$ be a simple connected graph of order n with m edges. If each edge of G belongs to at least t triangles ($t\ge 1$), then

${\lambda }_{n}\le max\left\{\frac{2d\left({v}_{i}\right)-t+\sqrt{4d\left({v}_{i}\right)m\left({v}_{i}\right)-4td\left({v}_{i}\right)+{t}^{2}}}{2d\left({v}_{i}\right)}:{v}_{i}\in V\right\},$
(3.3)

the equality occurs if G is the complete graph ${K}_{t+2}$.

Proof For $G={K}_{t+2}$, it is easy to check that the equality in Eq. (3.3) holds. If we replace Eq. (3.2) in the proof of Theorem 3.1 by

$\sum _{i=1}^{n}\underset{{v}_{j},{v}_{k}\in N\left({v}_{i}\right)}{\sum _{1\le j

then the result follows. □

For the maximal planar graphs, we have the following upper bound.

Theorem 3.3 Let G be a maximal planar graph of order $n\ge 4$ with m edges. Then

${\lambda }_{n}\le max\left\{\frac{d\left({v}_{i}\right)-1+\sqrt{d\left({v}_{i}\right)m\left({v}_{i}\right)-2d\left({v}_{i}\right)+1}}{d\left({v}_{i}\right)}:{v}_{i}\in V\left(G\right)\right\}.$
(3.4)

Proof Note that for any maximal planar graph G, each edge of G belongs to at least 2 triangles. Then the result follows from Theorem 3.2. □

In what follows, we turn to some lower bounds on ${\lambda }_{n}$. The following result due to Chung  concerns the lower bound on ${\lambda }_{n}\left(G\right)$.

Lemma 3.4 ()

Let G be a connected graph of order n. Then ${\lambda }_{n}\left(G\right)\ge \frac{n}{n-1}$, the equality holds if and only if $G\cong {K}_{n}$, where ${K}_{n}$ is the complete graph of order n.

Let $G=\left(V,E\right)$ be a graph and $X\subseteq V$ be a subset of the vertices. Let $\overline{X}=V\setminus X$ be the complement of the set X. The volume of X is defined to be the sum of the degrees of the vertices in G, that is,

$vol\left(X\right)=\sum _{v\in X}d\left(v\right).$

Note that $vol\left(V\right)$ is equal to twice the number of edges in the graph.

Theorem 3.5 Let G be a connected graph of order n with m edges. For any nonempty subset $X\subseteq V$, we have

${\lambda }_{n}\ge \frac{2m|{E}_{X}|}{vol\left(X\right)\left(2m-vol\left(X\right)\right)},$

where ${E}_{X}$ is the set of all edges with one end in X and the other end in $\overline{X}$. Moreover, if the equality holds, then $\frac{{\sum }_{u\in N\left(v\right)}f\left(u\right)}{d\left(v\right)}=x$ for each $v\in X$ and $\frac{{\sum }_{u\in N\left(v\right)}f\left(u\right)}{d\left(v\right)}=y$ for each $v\in \overline{X}$, where x and y are constant such that $\frac{x}{y}=-\frac{vol\left(\overline{X}\right)}{vol\left(X\right)}$.

Proof Let $X\subseteq V$ and f be a vector such that

Clearly, ${\sum }_{u\in V}d\left(u\right)f\left(u\right)=-vol\left(\overline{X}\right)vol\left(X\right)+vol\left(\overline{X}\right)vol\left(X\right)=0$. Moreover, note that $vol\left(X\right)+vol\left(\overline{X}\right)=vol\left(V\right)=2m$. Then, by Eq. (2.1), we have

$\begin{array}{rl}{\lambda }_{n}& \ge \frac{{\sum }_{uv\in E\left(G\right)}{\left(f\left(u\right)-f\left(v\right)\right)}^{2}}{{\sum }_{v\in V\left(G\right)}d\left(v\right)f{\left(v\right)}^{2}}\\ =\frac{|{E}_{X}|{\left(vol\left(X\right)+vol\left(\overline{X}\right)\right)}^{2}}{vol\left(X\right)vol\left(\overline{X}\right)\left(vol\left(X\right)+vol\left(\overline{X}\right)\right)}\\ =\frac{2m|{E}_{X}|}{vol\left(X\right)\left(2m-vol\left(X\right)\right)}.\end{array}$

Moreover, if the equality holds, then f is the harmonic eigenfunction of associated with ${\lambda }_{n}\left(G\right)$. Hence Proposition 2.1 implies that

Let $x=\left({\lambda }_{n}-1\right)vol\left(\overline{X}\right)$ and $y=\left(1-{\lambda }_{n}\right)vol\left(X\right)$. Then $\frac{x}{y}=-\frac{vol\left(\overline{X}\right)}{vol\left(X\right)}$. This completes the proof. □

Let $X=\left\{u\right\}$ in Theorem 3.5. Note that $vol\left(X\right)=d\left(u\right)=|{E}_{X}|$. Then we have the following.

Corollary 3.6 Let G be a graph of order n with m edges. Then

${\lambda }_{n}\ge \frac{2m}{2m-\mathrm{\Delta }},$
(3.5)

where Δ is the maximum degree of G.

Remark 3.1 Note that $2m\le n\mathrm{\Delta }$ holds for any graph of order n with m edges and maximum degree Δ. Thus the lower bound in Corollary 3.6 is always better than that in Lemma 3.4. Moreover, if G is a complete graph ${K}_{n}$ or a star ${S}_{n}$, then it is easy to check that the equality holds in Eq. (3.5).

Similarly, let $X=\left\{u,v\right\}$ in Theorem 3.5. Then we have:

Corollary 3.7 Let G be a graph of order n with m edges. Let $a={max}_{uv\in E\left(G\right)}\left\{d\left(u\right)+d\left(v\right)\right\}$ and $b={max}_{uv\notin E\left(G\right)}\left\{d\left(u\right)+d\left(v\right)\right\}$. Then:

1. (1)

${\lambda }_{n}\ge \frac{2m\left(a-2\right)}{a\left(2m-a\right)}$, and the equality holds if $G\cong {K}_{n}$.

2. (2)

${\lambda }_{n}\ge \frac{2m}{2m-b}$, and the equality holds if $G\cong {K}_{2,n-2}$, where $G\cong {K}_{2,n-2}$ is the complete bipartite graph with parts of cardinalities 2 and $n-2$.

Proof Let $X=\left\{u,v\right\}$ in Theorem 3.5. If $uv\in E\left(G\right)$, then $|{E}_{X}|=d\left(u\right)+d\left(v\right)-2$ and $vol\left(X\right)=d\left(u\right)+d\left(v\right)$. Theorem 3.5 implies that

${\lambda }_{n}\ge \frac{2m\left[\left(d\left(u\right)+d\left(v\right)\right)-2\right]}{\left(d\left(u\right)+d\left(v\right)\right)\left[2m-\left(d\left(u\right)+d\left(v\right)\right)\right]}.$

Let $f\left(x\right)=\frac{2m\left(x-2\right)}{x\left(2m-x\right)}$ for $x>2$. Then it is easy to see that $f\left(x\right)$ is increasing on x. Hence, we have ${\lambda }_{n}\ge \frac{2m\left(a-2\right)}{a\left(2m-a\right)}$. Moreover, it is easy to check that the equality holds when $G\cong {K}_{n}$.

If $uv\notin E\left(G\right)$, then $|{E}_{X}|=vol\left(X\right)=d\left(u\right)+d\left(v\right)$. Theorem 3.5 implies that

${\lambda }_{n}\ge \frac{2m}{2m-\left(d\left(u\right)+d\left(v\right)\right)}.$

Hence ${\lambda }_{n}\ge \frac{2m}{2m-b}$. Moreover, it is easy to check that the equality holds when $G\cong {K}_{2,n-2}$. □

A set of vertices X of G is called a cover of G if every edge of G is incident to some vertex in X. The least cardinality of a cover of G is called the covering number of G and denoted by $\tau \left(G\right)$. It is clear that if a vertex set X is a vertex cover if and only if $\overline{X}$ is an independent set. The following lower bound for ${\lambda }_{n}$ in terms of $\tau \left(G\right)$ is obtained.

Theorem 3.8 Let G be a graph order n with m edges. Then

${\lambda }_{n}\ge \frac{2m}{2m-\delta \left(n-\tau \left(G\right)\right)},$

where δ is the minimum degree of G. Moreover, the equality holds if $G\cong {C}_{n}$ when n is even, $G\cong {K}_{a,b}$ or $G\cong {K}_{n}$, where ${C}_{n}$ is the cycle of order n and ${K}_{a,b}$ is the complete bipartite graph with parts of cardinalities a and b.

Proof Let X be a minimal covering set of G with $|X|=\tau \left(G\right)$. Then $\overline{X}$ is an independent set. Hence $vol\left(\overline{X}\right)=|{E}_{X}|$ and $vol\left(X\right)=2m-|{E}_{X}|$. Then Theorem 3.5 implies that ${\lambda }_{n}\ge \frac{2m}{2m-|{E}_{X}|}$. Moreover, by the definition of covering set, we have $|{E}_{X}|\ge \delta \left(n-\tau \left(G\right)\right)$. Hence we have ${\lambda }_{n}\ge \frac{2m}{2m-\delta \left(n-\tau \left(G\right)\right)}$. Moreover, if $G\cong {C}_{n}$ when n is even, then $\tau \left(G\right)=\frac{n}{2}$. Hence it is easy to check that the equality holds. Similarly, if $G\cong {K}_{a,b}$ or $G\cong {K}_{n}$, then the equality holds. This completes the proof. □

Chung  proved that for any graph G of order n, ${\lambda }_{2}\le \frac{n}{n-1}$ with equality holding if and only if $G\cong {K}_{n}$. Moreover, the following result is also introduced.

Lemma 3.9 ()

Let G ($G\ne {K}_{n}$) be a connected graph of order n. Then ${\lambda }_{2}\le 1$.

In what follows, we characterize all connected graphs with ${\lambda }_{2}=1$. We will make use of the following lemma.

Lemma 3.10 ()

Let G be a connected graph of order n with maximum degree Δ and minimum degree δ. Let ${\rho }_{1}\le {\rho }_{2}\le \cdots \le {\rho }_{n}$ are the eigenvalues of $A\left(G\right)$. Then for each $1\le k\le n$,

$\frac{|{\rho }_{n-k+1}|}{\mathrm{\Delta }}\le |1-{\lambda }_{k}|\le \frac{|{\rho }_{n-k+1}|}{\delta }.$

Theorem 3.11 Let G ($G\ne {K}_{n}$) be a connected graph of order n. Then ${\lambda }_{2}=1$ if and only if G is a complete multipartite graph.

Proof By Lemma 3.10, if ${\lambda }_{2}=1$, then ${\rho }_{n-1}=0$, where ${\rho }_{n-1}$ is the second largest eigenvalue of $A\left(G\right)$. Hence the result follows from the fact that for any simple connected graph G of order n, ${\rho }_{n-1}\le 0$ if and only if G is a complete multipartite graph . On the other hand, when G ($G\ne {K}_{n}$) is a complete multipartite graph, ${\rho }_{n-1}\left(G\right)=0$ . This together with Lemma 3.10 imply that ${\lambda }_{2}=1$. The proof is completed. □

Moreover, the following result on ${\lambda }_{n-1}$ is also obtained.

Theorem 3.12 Let G be a connected graph of order n. Then ${\lambda }_{n-1}\ge 1$, the equality holds if and only if G is a complete bipartite graph.

Proof Note that for any connected graph of order n, ${\lambda }_{1}=0$ and ${\lambda }_{n}\le 2$. Since ${\sum }_{i=1}^{n}{\lambda }_{i}=n$, ${\sum }_{i=2}^{n-1}{\lambda }_{i}\ge n-2$ and hence ${\lambda }_{n-1}\ge 1$. Moreover, if ${\lambda }_{n-1}=1$, then ${\lambda }_{2}=\cdots ={\lambda }_{n-1}=1$ and ${\lambda }_{n}=2$ since ${\sum }_{i=2}^{n}{\lambda }_{i}=n$. This implies that G is bipartite . Moreover, since ${\lambda }_{2}=1$, combining with Theorem 3.11 we find that G is complete bipartite graph. On the other hand, it is easy to check that if G is a complete bipartite graph, then ${\lambda }_{n-1}=1$. This completes the proof. □

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## Acknowledgements

The authors would like to thank the anonymous referees for their constructive corrections and valuable comments on this paper, which have considerably improved the presentation of this paper. This work was supported by NSF of China (Nos. 11101358, 61379021, 11371372); NSF of Fujian (No. 2014J01020); China Postdoctoral Science Foundation (No. 2014M551831); General Research Fund of Hong Kong; Faculty Research Grant of Hong Kong Baptist University.

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Correspondence to Jianxi Li.

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The authors declare that they have no competing interests.

### Authors’ contributions

JL carried out the proofs of the main results in the manuscript. J-MG and WCS participated in the design of the study and drafted the manuscript. All authors read and approved the final manuscript.

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