On the eigenvalues of weighted directed graphs

TL;DR

This paper explores the spectral properties of weighted directed graphs by introducing a self-adjoint operator and analyzing how graph perturbations influence eigenvalues, extending matrix analysis techniques to graph Laplacians.

Contribution

It introduces a new self-adjoint operator for weighted directed graphs and generalizes matrix analysis methods to study eigenvalue behavior under graph perturbations.

Findings

Eigenvalues are affected by graph perturbations.

A new self-adjoint operator is proposed for spectral analysis.

Techniques from matrix analysis are extended to directed graph Laplacians.

Abstract

This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as the sum of two non self-adjoint Laplacians. We investigate how the perturbation of the graph can affect the eigenvalues. Our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized for graph Laplacians.

Figures

Equations

$$v(x)=\#V_x\text{leavevmode}\text{leavevmode}\text{ for all }x\in V.$$

$$x_1=x,\text{leavevmode}{y}_n=y\text{ and }{x}_i=y_{i-1}\text{leavevmode}\text{leavevmode}\forall \text{leavevmode}2\le i\le n.$$

$$(0,1)\in \vec{E},\text{leavevmode}(1,2)\in \vec{E},..,(n-2,n-1)\in \vec{E},\text{leavevmode}(n-1,0)\in \vec{E}$$

$$\overset{\circ}{\Omega}=\big{\{}y\in\Omega,\leavevmode\nobreak\ V_{y}\subset\Omega\big{\}}$$

$$\partial\Omega=\big{\{}y\in\Omega^{c},\leavevmode\nobreak\ y\in V_{x}\leavevmode\nobreak\ \text{ for some }x\in\Omega\big{\}}$$

$$\partial_{E}\Omega=\big{\{}(x,y)\in\vec{E}:\leavevmode\nobreak\ (x\in\Omega,\leavevmode\nobreak\ y\in\Omega^{c})\leavevmode\nobreak\ \leavevmode\nobreak\ or\leavevmode\nobreak\ \leavevmode\nobreak\ (x\in\Omega^{c},\leavevmode\nobreak\ y\in\Omega)\big{\}}.$$

$$\forall \text{leavevmode}x\in V,\text{leavevmode}\text{leavevmode}\#V_x^{+}\ne 0\text{leavevmode}and\text{leavevmode}\#V_x^{-}\ne 0.$$

$${\beta }^{+}(x)=\sum _{y\in V_x^{+}}b(x,y)\text{ and }{\beta }^{-}(x)=\sum _{y\in V_x^{-}}b(y,x).$$

$${\beta }_G(x)={\beta }^{+}(x)+{\beta }^{-}(x)=2{\beta }^{+}(x).$$

$$m:V\to {\mathbf{R}}_{+}^{\ast }.$$

$${\mathcal{C}}_m(V)=\{f:V\to \mathbb{C}\}$$

$$(f,g{)}_m=\sum _{x\in V}m(x)f(x)\overline{g(x)}.$$

$$\Vert f{\Vert }_m=\sqrt{(f,f{)}_m}.$$

$${\mathcal{C}}_m(U)=\{f\in {\mathcal{C}}_m(V),\text{leavevmode}f\text{ with support in U}\}.$$

$$\Delta_{G}f(x)=\frac{1}{m(x)}\sum_{y\in V_{x}^{+}}b(x,y)\big{(}f(x)-f(y)\big{)}.$$

$$\tilde{\Delta}_{G}f(x)=\frac{1}{\beta^{+}(x)}\sum_{y\in V^{+}_{x}}b(x,y)\big{(}f(x)-f(y)\big{)}.$$

$$f\text{ is with support in }U,\text{leavevmode}\text{leavevmode}{A}_U^D(f)=A(f){|}_U.$$

$$\forall \varphi ,\psi \in {\mathcal{C}}_m(V),\text{leavevmode}(\Delta \psi ,\varphi )=(\psi ,{\Delta }^{\ast }\varphi ).$$

$$\Delta^{*}_{G}f(x)=\frac{1}{m(x)}\sum_{y\in V_{x}^{-}}b(y,x)\big{(}f(x)-f(y)\big{)}.$$

$$({\Delta }_{G}f,g{)}_m=$$

$$=$$

$$(\Delta_{G}f,g)_{m}+(\Delta^{*}_{G}f,g)_{m}=\sum_{(x,y)\in\vec{E}}b(x,y)\big{(}f(x)-f(y)\big{)}\big{(}\overline{g(x)-g(y)}\big{)}.$$

$$({\Delta }_{G}f,g{)}_m+({\Delta }_G^{\ast }f,g{)}_m=$$

$$+$$

$$=$$

$$S_{G}f(x)$$

$$\displaystyle=\frac{1}{m(x)}\sum_{y\in V_{x}^{+}\cup V_{x}^{-}}\big{(}b(x,y)+b(y,x)\big{)}\big{(}f(x)-f(y)\big{)}$$

$$\displaystyle=\frac{1}{m(x)}\sum_{y\in V_{x}}a(x,y)\big{(}f(x)-f(y)\big{)}$$

$$(S_{G}f,f)=$$

$$=$$

$$\ge$$

$${\lambda }_1\le {\lambda }_2\le ...\le {\lambda }_n$$

Full text

1. 1 Preliminaries

2. 1.1 Notion of Graphs

3. 1.2 Functional spaces

4. 2 Laplacian on directed graphs

5. 2.1 Variational principles and Properties

6. 3 Domain monotonicity of eigenvalues

7. 3.1 Definitions on $G$

8. 3.2 Monotonicity relative to vertices

9. 3.3 Monotonicity relative to edges

10. 4 Comparison eigenvalues of Dirichlet Laplacian on graphs

Introduction

This article follows up on the ideas in [4] and [12] on monotonicity eigenvalues which are relative to continuous domains. The purpose of this work is to explore in the case of weighted directed graphs, some familiar facts of monotonicity proved on domains of $\mathbb{R}^{n}$ and on compact Riemannian manifolds. Specially, our main question is: ”can one study the behavior of the eigenvalues of a special operator under perturbations on finite graphs?” First, we consider a finite, directed and connected graph $G$ with non symmetric edge weights. Then, we introduce the associated non symmetric Laplacian $\Delta_{G}$. We introduce the self-adjoint operator $S_{G}=\Delta_{G}+\Delta^{*}_{G}$, such that it is easier to examine its spectrum thanks to selfadjointness. We give some spectral properties of $\Delta_{G}$ and we show that the real part of its eigenvalues can coincide with the eigenvalues of $\dfrac{1}{2}S_{G}$. Secondly, we study the monotonicity of eigenvalues relative to vertices or edges of $G$. We prove that the $k^{th}$ eigenvalue $\lambda_{k}$ of $S_{G}$ is decreasing only in a class of graphs called here flower-like-graphs, and it is monotone increasing in the set of edges. These results are inspired by the classical results going back from M. Fiedler [6] and P. Kurasov, G. Malenova, S. Naboko [11] on the first nonzero eigenvalue of a simple graph. We extend these results for the higher eigenvalues $\lambda_{k}$ of our special operator. In the second part of our work, we try to establish and improve the Proposition 2.1 of [3] for a Riemannian manifold $M$ which gives upper bounds on the higher eigenvalues $\lambda_{k}(M)$ in terms of Dirichlet eigenvalues on components of a partition of $M$.

Let us briefly outline the contents of this article. We shall start with a short section of preliminaries consisting on some basic properties of the non symmetric Laplacian $\Delta_{G}$ on $G$ and the associated Green formula. In section 3, we establish a generalization of some monotonicity eigenvalue results. Furthermore, we study the example of flower-like graphs and insist on the interest of the subgraph concept. Hence, we can remark that these considerations of graph help to give an upper bound of the eigenvalues of a simple tree. This section includes also similar Weyl and Cauchy theorems for the matrices [10]. In section 4, we are interested on the study of the eigenvalues of the Dirichlet Laplacian. We involve a comparison on the eigenvalues to use the decomposition of $G$ into two components $A$ and $B$ and we give an upper bound on the eigenvalues of $G$ in terms of Dirichlet eigenvalues of $A$ and $B$.

1. Preliminaries

We will review in this section some basic definitions and introduce the notation used in the article. They are introduced in [2] for the infinite graph.

1.1. Notion of Graphs

We call oriented or directed graph, the couple $G=(V,\vec{E})$, where $V$ is a set of vertices, and $\vec{E}\subset V\times V$ is a set of directed edges. For two vertices $x,y$ of $V$, we denote by $(x,y)$ the edge that connects $x$ to $y$, we also say that $x$ and $y$ are * neighbors*.

For all $x\in V$, we set:

• •

$E=\left\{\{x,y\},\leavevmode\nobreak\ (x,y)\in\vec{E}\text{ or }(y,x)\in\vec{E}\right\}$

• •

$V_{x}^{+}=\left\{y\in V,\leavevmode\nobreak\ \leavevmode\nobreak\ (x,y)\in\vec{E}\right\}$

• •

$V_{x}^{-}=\left\{y\in V,\leavevmode\nobreak\ \leavevmode\nobreak\ (y,x)\in\vec{E}\right\}$

• •

$V_{x}=V_{x}^{+}\cup V_{x}^{-}$.

The valency on $G$ is given by:

[TABLE]

We introduce some definitions given in [1], [2], [5], [14] for the case of symmetric graphs.

Definitions 1.1**.**

• •

A path between two vertices $x$ and $y$ in $V$ is a finite set of directed edges $(x_{1},y_{1});\leavevmode\nobreak\ (x_{2},y_{2});..;(x_{n},y_{n}),\leavevmode\nobreak\ n\geq 1$ such that

[TABLE]

• •

$G=(V,\vec{E})$* is called connected if two vertices are always related by a path.*

• •

$G=(V,\vec{E})$* is called strongly connected if there is for all vertices $x,y$ a path from $x$ to $y$ and one from $y$ to $x$.*

Example 1.1**.**

The cycle graph $C_{n}=\{0,1,..,n-1\}$, with

[TABLE]

is strongly connected.

• •

Define for a finite subset $\Omega$ of  $V$, the interior, the vertex boundary and the edge boundary of $\Omega$ respectively by:

[TABLE]

[TABLE]

[TABLE]

We remark that the strong connectedness of $G$ assures that:

[TABLE]

In this work we suppose that $G$ is finite, connected and satisfies the Hypothesis (1). In this work, we take the following definition.

Definition 1.1**.**

Directed weighted Graph: A weighted graph $(G,b)$ is the data of a graph $(V,\vec{E})$ and a weight $b:V\times V\to\mathbb{R}_{+}$ satisfying the following conditions:

• •

$b(x,x)=0$* for all $x\in V$,   (no loops in $\vec{E}$)*

• •

$b(x,y)>0$* iff $(x,y)\in\vec{E}$*

• •

Assumption $(\beta)$.

Assumption $(\beta)$: for all $x\in V$,  $\beta^{+}(x)=\beta^{-}(x)$

where

[TABLE]

The weight $\beta_{G}$ on a vertex $x\in V$ is given by:

[TABLE]

Remark 1.1**.**

The Assumption $(\beta)$ is natural, it looks like the Kirchhoff’s law in the electrical networks.

The weighted graph is symmetric if for all $x,y\in V$, $b(x,y)=b(y,x)$,

as a consequence $(x,y)\in\vec{E}\Rightarrow(y,x)\in\vec{E}$ (the graph is symmetric).

In addition, we consider a weight $m$ on $V$:

[TABLE]

1.2. Functional spaces

Let us introduce the following function spaces associated to the graph $G$:

[TABLE]

endowed with the following inner product:

[TABLE]

We define its associated norm by:

[TABLE]

A particular case called normalized is for $m=\beta^{+}$.

For a subset $U$ of $V$, Let

[TABLE]

The weights $m$ and $b$ are called simple if they are constant equal to $1$ on $V$ and $E$ respectively. We denote by $G^{s}$ the simple graph (with simple weights).

2. Laplacian on directed graphs

For a weighted connected directed graph $(G,b)$, we introduce the combinatorial Laplacians:

Definitions 2.1**.**

• •

We define the Laplacian $\Delta_{G}$ on $\mathcal{C}_{m}(V)$ by:

[TABLE]

• •

In particular, if for all $x\in V$, $\beta^{+}(x)=m(x)$, the Laplacian is said to be the normalized Laplacian and is defined on $\mathcal{C}_{\beta^{+}}(V)$ by:

[TABLE]

• •

For any operator $A$ on $\mathcal{C}_{m}(V)$, the Dirichlet operator $A^{D}_{U}$, where $U$ is a subset of $V$, is defined by:

[TABLE]

Thanks to Hypothesis $(\beta)$ the adjoint of $\Delta$ has a simple expression.

Definition 2.1**.**

Adjoint of an operator: The adjoint operator $\Delta^{*}$ of $\Delta$ is defined by:

[TABLE]

Proposition 2.1**.**

Let $f$ be a function of $\mathcal{C}_{m}(V)$, we have

[TABLE]

Proof:

The following calculation for all $f,g\in\mathcal{C}_{m}(V)$ gives:

[TABLE]

$\square$

The Green’s formula is one of the main tools when we are working with the symmetric Laplace operator. In the following we establish it for the non symmetric Laplacian.

Lemma 2.1**.**

Green’s Formula. Let $f$ and $g$ be two functions of $\mathcal{C}_{m}(V)$. Then

[TABLE]

Proof:

The proof is a simple calculation:

[TABLE]

$\square$

Definition 2.2**.**

Special Laplacian. We define a special Laplacian $S_{G}$ as the sum of the two non self-adjoint Laplacians $\Delta_{G}$ and $\Delta^{*}_{G}$, given by:

[TABLE]

where $a(x,y)=b(x,y)+b(y,x)$ for any $x,y\in V$.

Remark 2.1**.**

1. (1)

$S_{G}$* is a symmetric operator on $\mathcal{C}_{m}(V)$, because $(\Delta^{*}_{G})^{*}=\Delta_{G}$.* 2. (2)

$S_{G}$* is a positive operator: for all $f\in\mathcal{C}_{m}(V)$,*

[TABLE]

In our discussion on the study of eigenvalues of a self-adjoint operator, it is natural to introduce the different characterization by variational principles [9].

2.1. Variational principles and Properties

Let $A$ be a bounded from below self-adjoint operator. The eigenvalues of $A$ can be characterized by three fundamental variational principles: the Rayleigh’s principle, the Poincaré-Ritz max-min principle and the Courant-Fischer-Weyl principle applied to the Rayleigh quotients $\mathcal{R}(f)=\dfrac{(Af,f)}{(f,f)}$, $f\neq 0$.

Let us arrange the eigenvalues of $A$ as

[TABLE]

counted according to their multiplicities.

In this case we have :

1. (1)

The Rayleigh’s principle states:

[TABLE]

where $f_{i}$ are eigenvectors corresponding to the eigenvalues $\lambda_{i}$ and the minimum is reached at the eigenvector $f_{k}$. 2. (2)

The Poincaré-Ritz principle establishes:

[TABLE] 3. (3)

The Courant-Fischer-Weyl principle is given in the form:

[TABLE]

The result below establishes a link between the eigenvalues of $S_{G}$ and $\Delta_{G}$. We assume that the eigenvalues of $\Delta_{G}$ are ordered as follows respectively:

[TABLE]

Lemma 2.2**.**

[TABLE]

Proof:

Let $f$ be an eigenfunction associated to $\lambda_{n}(\Delta_{G})$, we have

[TABLE]

$\square$

Remark 2.2**.**

In a particular case, the previous inequalities are strict. Let us consider the following example, where $m\equiv 1$,

we have $\sigma(S_{G})=\{0,3,7\}$ and $\sigma(\Delta_{G})=\{0,2,3\}$.

We introduce in the following a particular case of graphs whose \sigma(S_{G})=2\mathcal{R}e\big{(}\sigma(\Delta_{G})\big{)}.

Example 2.1**.**

Let us consider the simple cycle graph $C_{3}$, see the Figure 2, we have ,

[TABLE]

In the following proposition we determine the spectrum of the non symmetric Laplacian $\Delta_{C_{n}}$. We follow the same approach as Grigoryan for the symmetric Laplacian [7] page 49.

Proposition 2.2**.**

The eigenvalues of $\Delta_{C_{n}}=I-P$ are as follows:

1. (1)

If $n$ is odd then the eigenvalues are $\lambda=0$ (simple) and $\lambda=1-e^{\pm i\frac{2l\pi}{n}}$ for all $l=1,..,\frac{n-1}{2}$ (simple). 2. (2)

If $n$ is even then the eigenvalues are $\lambda=0,\leavevmode\nobreak\ 2$ (simple) and $\lambda=1-e^{\pm i\frac{2l\pi}{n}}$ for all $l=1,..,\frac{n}{2}-1$ (simple).

Proof:

To compute the eigenvalues of $\Delta_{C_{n}}=I-P$, it is sufficient to determine the spectrum of $P$. Let $\alpha$ be an eigenvalue of the operator $Pf(k+1)=f(k)$, for $k=0,..,n-1$, which leads to $f(k)=\alpha^{k}f(0)$ but $f(n)=f(0)$ thus $\alpha^{n}=1$. As $f$ is an eigenfunction $(f(0)\neq 0)$, then $\alpha=e^{\pm i\theta}$. As $f$ is n-periodic provided $n\theta$ is a multiple of $2\pi$, hence,

[TABLE]

where $l$ is an integer of $(0,\frac{n}{2})$.

$\square$

Observe that an interesting corollary concerning the spectra of $S_{C_{n}}$ and $\Delta_{C_{n}}$.

Corollary 2.1**.**

[TABLE]

Proof:

We refer the Lemma 2.7 of [7], we remark that the eigenvalues of the operator \dfrac{1}{2}(P+P^{*})f(k)=\dfrac{1}{2}\big{(}f(k+1)+f(k-1)\big{)} coincide with the real part of the eigenvalues of $P$.

$\square$

Using the Green’s formula, we establish some properties of the spectrum on any graph $G$ .

Proposition 2.3**.**

1. (1)

[math]* is a simple eigenvalue of $\tilde{S}_{G}$ and $S_{G}$.* 2. (2)

All the eigenvalues of $\tilde{S}_{G}$ are contained in $[0,4]$. 3. (3)

The real part of the eigenvalues of $\tilde{\Delta}_{G}$ are also contained in $[0,2]$.

Proof:

1. (1)

As in the case of undirected graph [7], we have for all $f\in\mathcal{C}_{\beta^{+}}(V)$,

[TABLE]

Clearly, the constant function is an eigenfunction of [math]. Assume now that f is an eigenfunction of the eigenvalue [math]. By the connectedness of $G$, $f$ is constant, which will imply that [math] is a simple eigenvalue. It is similar for $\tilde{S}_{G}$. 2. (2)

It is sufficient to prove that $\tilde{S}_{G}$ is bounded by $4$ because $\tilde{S}_{G}$ is non negative by the Green’s formula. In fact, for all $f\in\mathcal{C}_{\beta^{+}}(V)$ and thanks to Assumption $(\beta)$, we obtain

[TABLE] 3. (3)

We deduce directly our inclusion thanks to the Lemma 2.2.

$\square$

3. Domain monotonicity of eigenvalues

The purpose of this part is to give an overview of some results concerning the monotonicity with regard to the domain, of eigenvalues of $S_{G}$, the special self-adjoint Laplacian associated to directed graphs with non symmetric edge weights. We could be concerned with the related question:

Does a given eigenvalue increases or decreases under a given perturbation of $G$?

3.1. Definitions on $G$

Before discussing the study of variation of eigenvalues, let us recall some basic definitions: let $G=(V,\vec{E})$ be a graph,

• •

The graph $G_{1}=(V,\vec{E}_{1})$ is called a partial graph of $G$, if $\vec{E}_{1}$ is included in $\vec{E}$.

• •

A graph $H=(V_{H},\vec{E}_{H})$ is called a subgraph of $G=(V_{G},\vec{E}_{G})$ if $V_{H}\subset V_{G}$ and \vec{E}_{H}=\big{\{}(x,y);\leavevmode\nobreak\ x,y\in V_{H}\leavevmode\nobreak\ \leavevmode\nobreak\ \big{\}}\cap\vec{E}_{G}.

• •

A graph $(V_{U},\vec{E}_{U})$ is called a part of a graph $G=(V_{G},\vec{E}_{G})$ if $V_{U}\subset V_{G}$ and \vec{E}_{U}=\big{\{}(x,y),\leavevmode\nobreak\ \leavevmode\nobreak\ x,y\in V_{U}\big{\}}\subset\vec{E}_{G}.

Remark 3.1**.**

A subgraph is a part of $G$ but the converse is not true, for example let us give the following undirected graphs, see the Figure 3.

$H$* is a part of $G$, but not a subgraph.*

Remark 3.2**.**

A confusion between a subgraph and a part of a graph can create a false interpretation on the monotonicity of eigenvalue.

3.2. Monotonicity relative to vertices

We study the monotonicity of eigenvalues under the variation of the set of vertices.

By the Courant-Fischer-Weyl principle we establish the following statement.

Theorem 3.1**.**

Let $H=(V_{H},\vec{E}_{H})$ be a connected subgraph of a graph $G=(V_{G},\vec{E}_{G})$, $\sharp V_{G}=n$, then for any $1\leq k\leq\sharp V_{H}=r$ :

[TABLE]

Proof:

Let $f_{1},\leavevmode\nobreak\ f_{2},...,f_{k}$ be $k$ eigenfunctions associated to $\lambda_{1}(S_{H}),\leavevmode\nobreak\ \lambda_{2}(S_{H}),...,\lambda_{k}(S_{H})$; and $F=\{f_{1},\leavevmode\nobreak\ f_{2},...,f_{k},\delta_{1},..,\delta_{n-r}\}$, $\delta_{1},..,\delta_{n-r}$ are the Dirac measures on $G$ relative to the vertices in $V_{G}\setminus V_{H}$. It is clear that $\dim F=k+n-r$. Then using (4), we obtain ;

[TABLE]

hence $\exists\varphi_{k}\in F^{\perp}$ with support in $H$ such that

[TABLE]

$\square$

For studying the behavior of eigenvalues relative to perturbations, we propose a special construction of graphs.

Definition 3.1**.**

Let $G=(V_{G},\vec{E}_{G})$ be a weighted graph. $G$ is called $H$-flower-like with respect to the subgraph $H=(V_{H},\vec{E}_{H})$ of $G$ if there exists $(H_{i})_{i\in I}$ a family of subgraphs of $G$ such that:

1. (1)

$V_{G}=V_{H}\cup(\displaystyle{\uplus_{i}V_{\overset{\circ}{{H_{i}}}}})$** 2. (2)

$\forall\leavevmode\nobreak\ i,j\in I,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ x\in V_{\overset{\circ}{{H_{i}}}},\leavevmode\nobreak\ \leavevmode\nobreak\ y\in V_{\overset{\circ}{{H_{j}}}},\leavevmode\nobreak\ i\neq j\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ x\notin V_{y}$. 3. (3)

$\forall\leavevmode\nobreak\ i\in I,\leavevmode\nobreak\ \exists\leavevmode\nobreak\ x_{i}\in V_{G},\leavevmode\nobreak\ V_{H}\cap V_{H_{i}}=\{x_{i}\}$.

These special graphs are used to create a rule of monotonicity of $\lambda_{k}$ under a given graph perturbation.

Theorem 3.2**.**

Let $G=(V_{G},\vec{E}_{G})$ be a $H$-flower-like graph, we have then for any $1\leq k\leq\sharp V_{H}=r$,

[TABLE]

Proof:

We use the variational principle (2). Let be $f_{1},\leavevmode\nobreak\ f_{2},...,f_{k-1}$ eigenfunctions of $G$ in $\mathcal{C}_{m}(V)$ associated to $\lambda_{1}(S_{G}),\leavevmode\nobreak\ \lambda_{2}(S_{G}),...,\lambda_{k-1}(S_{G})$, and $g_{j}$ be the eigenfunction associated to $\lambda_{j}(S_{H})$, for $j=1,..,k$. We define a function $\phi_{j}$ on $V_{G}$ by:

[TABLE]

where $\{x_{s}\}=V_{H}\cap V_{H_{s}},\leavevmode\nobreak\ \forall s\in I=\{1,..,r\}$

Let $F=Span\{\phi_{j}\}_{1\leq j\leq k}$, $\dim F=k$ so, there exist $k$ reals $\alpha_{1},\alpha_{2},...,\alpha_{k}$ not all equal to zero, satisfying:

[TABLE]

Therefore the function $\psi(x)=\displaystyle{\sum_{j=1}^{k}}\alpha_{j}\phi_{j}(x)\neq 0$ is orthogonal to $f_{l}$ for all $1\leq l\leq k-1$, we define for $x\in H$, $\psi^{\prime}(x)=\displaystyle{\sum_{j=1}^{k}}\alpha_{j}g_{j}(x)$. Then we get :

[TABLE]

as

[TABLE]

then

[TABLE]

$\square$

The above results have several important consequences, for instance on a tree seen as a flower-like graph.

Corollary 3.1**.**

Let $G^{s}$ be a simple symmetric tree and $2q=\displaystyle{\max_{x\in V}v(x)}$ then the eigenvalues of $G^{s}$ satisfy for all $k=1,..,q$:

[TABLE]

and

[TABLE]

Proof:

Clearly there exists a symmetric star graph $S_{q}$ with $q+1$ vertices seen as a subgraph of $G^{s}$. It can be considered as a $S_{q}$-flower-like graph. Therefore $G^{s}$ satisfies the assumptions of the Proposition 3.2. Hence we get the result because the spectrum of $S_{q}$ is : $0,2,...,2,2(q+1)$, see [13].

$\square$

In the following, we will show more general results : instead of adding only one vertex and one edge, we would also add a graph.

Corollary 3.2**.**

Let $G$ be a graph with $n$ vertices, let $G_{1}$ connected to G by a single edge.Then for $k=1,..,n$;

[TABLE]

Remark 3.3**.**

The previous corollary is an immediate consequence of the Proposition 3.2. In addition this is an interesting generalization of Proposition 2 in [11] which shows that $\lambda_{2}(S_{G})\geq\lambda_{2}(S_{G_{1}})$ where $G_{1}$ is obtained from $G$ by adding one edge between one vertex of $G$ and one new vertex.

3.3. Monotonicity relative to edges

We apply the Weyl and interlacing Theorems for matrices to study how the spectrum of the special operator $S_{G}$ of a directed graph changes under adding an edge or a set of edges.

Now we have several opportunities to refer to the following basic observation about subspace intersections (see [10] page 235).

Lemma 3.1**.**

Let $W$ be a finite dimensional vector space and let $S_{1}$, $S_{2}$,..,$S_{k}$ be subspaces of $W$, if

[TABLE]

then  $\dim(S_{1}\cap..\cap S_{k})\geq\delta\text{ and hence }S_{1}\cap..\cap S_{k}\text{ contains }\delta$ linearly independent vectors, in particular , it contains a nonzero vector.

Observation 3.1**.**

Let $S_{G}$ be the special self-adjoint operator with eigenvalues $\lambda_{1}(S_{G})\leq\lambda_{2}(S_{G})\leq...\leq\lambda_{n}(S_{G})$. Then the ordered eigenvalues of $-S$ are $\lambda_{1}(-S_{G})\leq\lambda_{2}(-S_{G})\leq...\leq\lambda_{n}(-S_{G})$, that is, $\lambda_{k}(-S_{G})=-\lambda_{n-k+1}(S_{G}),\leavevmode\nobreak\ k=1,...,n.$

We show that the $k^{th}$ eigenvalue $\lambda_{k}(S_{G})$ is monotonously increasing functions of the set of edges. The following results are the generalizations of the Proposition 1 and the Proposition 2 in [11].

Proposition 3.1**.**

Let $G=(V,\vec{E})$ be a connected finite weighted graph $(\sharp V=n)$, consider the partial graphs $G_{1}=(V,\vec{E}_{1})$ and $G_{2}=(V,\vec{E}_{2})$ where $E=E_{1}\sqcup E_{2}$ (disjoint union). Then for all $k=1,...,n$ and $r,s=1,2$, $r\neq s$:

[TABLE]

and

[TABLE]

Proof:

Let $g_{k}$,  $h_{k}$, and $f_{k}$ be the eigenfunctions associated to $\lambda_{k}(S_{G_{1}})$, $\lambda_{k}(S_{G_{2}})$ and $\lambda_{k}(S_{G})$ respectively for $k=1,..,n$.

For $j=0,...,n-k$, we define $S_{1}=Span\{g_{1},...,g_{k+j}\}$, $S_{2}=Span\{h_{1},...,h_{n-j}\}$ and $S_{3}=Span\{f_{k},...,f_{n}\}$ by the Lemma (3.1) there exists a non zero function $\psi$ in $S_{1}\cap S_{2}\cap S_{3}$ and so we will have

[TABLE]

In the following we apply the equality (6) to the operator $-S_{G}$ because the inequalities (7) does not depend on positivity of $b$, and by:

[TABLE]

We obtain by re-indexing:

[TABLE]

$\square$

We can easily deduce,

Corollary 3.3**.**

Let $G=(V,\vec{E})$ be a connected finite graph, $G_{1}=(V,\vec{E}_{1})$ and $G_{2}=(V,\vec{E}_{2})$ two partial graphs of $G$ where $\sharp V=n\text{ and }E=E_{1}\sqcup E_{2}$, then for all $k=1,...,n$ and $r,s=1,2$, $r\neq s$:

[TABLE]

Proof:

By applying the proposition 3.1 to $j=0$ and $j=1$ respectively, we obtain the result because $\lambda_{1}(S_{G_{r}})=0.$

$\square$

In other words, adding a subset of edges to $\vec{E}$ while keeping the same set of vertices always induces an increasing of the $k^{th}$ eigenvalue or keeps it unchanged.

Corollary 3.4**.**

Let $G=(V,\vec{E})$ be a connected finite graph with $n$ vertices, and $G_{1}$ a graph obtained by adding a set of edges to $G$ then for all $k=1,...,n$:

[TABLE]

4. Comparison eigenvalues of Dirichlet Laplacian on graphs

In this section, we present some results about the spectrum comparison between the Laplacian and the Dirichlet Laplacian. The purpose of this part is to find the relation between the usual vertex weight on a subgraph $H$ of $G$ and its boundary weight to compare eigenvalues.

This is done by establishing a clear and explicit link between the eigenvalues and the Dirichlet eigenvalues on $G$.

In the following proposition, we treat the Dirichlet Laplacian case :

[TABLE]

By the same techniques used in the Lemma 2.2, we can show the following Lemma.

Lemma 4.1**.**

[TABLE]

and

[TABLE]

Proof:

Let $f$ and $g$ be eigenfunctions associated to $\lambda_{1}(\Delta^{D}_{U})$ and $\lambda_{n}(\Delta^{D}_{U})$ respectively. By the variational principle of $\lambda_{1}(S^{D}_{U})$ and $\lambda_{n}(S^{D}_{U})$, we have

[TABLE]

and

[TABLE]

$\square$

In the same spirit as the Cauchy interlacing theorem concerning hermitian bordered matrices (see [9] theorem 4.3.28 for a generalized statement) one can prove the following.

Proposition 4.1**.**

Consider a connected subgraph $H=(V_{H},\vec{E}_{H})$ of $G=(V_{G},\vec{E}_{G})$, $(\#V_{G}=n,\#V_{H}=r)$, then the eigenvalues on $G$ of the normalized Laplacian $S_{G}$ satisfies:

[TABLE]

Proof:

Let $h_{1},..,h_{r}$ and $f_{1},..,f_{n}$ be the eigenfunctions associated to $S^{D}_{H}$ and $S_{G}$ respectively, define the function $g_{k}$ for $k=1,..,r$ by:

[TABLE]

Let $1\leq k\leq r$ and fix $S_{1}=Span\{g_{k},...,g_{r}\}$ and $S_{2}=Span\{f_{1},...,f_{k+n-r}\}$ by the Lemma 3.1 there exists a function $\psi$ in $S_{1}\cap S_{2}$. Since $\psi\in S_{1}$, it has the form

[TABLE]

for some $g\in Span\{g_{k},...,g_{r}\}$. Observe that:

[TABLE]

since $(g,g)_{m}=(\psi,\psi)_{m}$, we get

[TABLE]

then,

[TABLE]

$\square$

We deduce easily from the Proposition 4.1 an estimation of the eigenvalues of $S_{G}$ and $\Delta_{G}$, thanks to Theorem 3.1, as follows:

Corollary 4.1**.**

Consider a connected subgraph $H=(V_{H},\vec{E}_{H})$ of $G=(V_{G},\vec{E}_{G})$, $(\#V_{G}=n,\#V_{H}=r)$, then the eigenvalues on $G$ satisfies:

[TABLE]

Corollary 4.2**.**

Consider a connected subgraph $H=(V_{H},\vec{E}_{H})$ of $G=(V_{G},\vec{E}_{G})$, $(\#V_{G}=n,\#V_{H}=r)$, then:

[TABLE]

Corollary 4.3**.**

Consider a connected subgraph $H=(V_{H},\vec{E}_{H})$ of the cycle graph $C_{n}$, $(\#V_{H}=r)$, then:

[TABLE]

In the following Proposition we prove how to give an upper bound for $\lambda_{k+l}(S_{G})$ in terms of a Rayleigh quotient. We give a discrete version of the Proposition 2.1 [3] by applying the Poincaré min-max principle. The methods we use follow closely the arguments given in B. Benson [3] in the case of the Laplacian of Riemannian manifolds. Next we provide an upper bound of the eigenvalues of $G$ according to Dirichlet eigenvalues on such repartition of $G$ as in the Figure 5.

Proposition 4.2**.**

Let $G=(V,\vec{E})$ a finite connected graph, $U=(V_{U},\vec{E}_{U})$ a part of $G$ and $A=(V_{A},\vec{E}_{A})$, $B=(V_{B},\vec{E}_{B})$ two subgraphs satisfying the following conditions:

1. (1)

$V=V_{A}\sqcup V_{B}=V_{\overset{\circ}{A}}\sqcup V_{\overset{\circ}{B}}\sqcup V_{U}$*    (** disjoint union**)* 2. (2)

$\vec{E}=\vec{E}_{A}\sqcup\vec{E}_{B}\sqcup\vec{E}_{U}$** 3. (3)

$\partial_{\vec{E}}A=\partial_{\vec{E}}B=\vec{E}_{U}$.

Then we get for $1\leq k,l\leq\min(\sharp V_{\overset{\circ}{A}},\sharp V_{\overset{\circ}{B}})$:

[TABLE]

Proof:

Let $h_{k}$ and $g_{l}$ be the eigenvectors associated to $\lambda_{k}(S^{D}_{\overset{\circ}{A}})$ and $\lambda_{l}(S^{D}_{\overset{\circ}{B}})$ respectively, for $1\leq k\leq\sharp V_{\overset{\circ}{A}}$ and $1\leq l\leq\sharp V_{\overset{\circ}{B}}$.

We define $f$ on $\mathcal{C}_{m}(\overset{\circ}{A}\oplus\overset{\circ}{B})$ by:

[TABLE]

We have

[TABLE]

Using the Poincaré min-max principle (3), we obtain:

[TABLE]

Hence

[TABLE]

$\square$

Remark 4.1**.**

The previous Proposition remains true in the particular case of the Laplacian $\tilde{S}_{G}$.

An estimate of $\lambda_{2}(S_{G})$ can also be obtained with respect to $\lambda_{1}(\Delta^{D}_{\overset{\circ}{B}})$ and $\lambda_{1}(\Delta^{D}_{\overset{\circ}{B}})$.

Corollary 4.4**.**

Under the same hypothesis of the previous Proposition we have

[TABLE]

Corollary 4.5**.**

Under the same hypothesis of the previous Proposition, from the cycle graph $C_{n}$, we have

[TABLE]

Acknowledgement: I take this opportunity to express my gratitude to my PhD advisors Colette Anné and Nabila Torki-Hamza for all the fruitful discussions, helpful suggestions and their guidance during this work. This work was financially supported by the ”PHC Utique” program of the French Ministry of Foreign Affairs and Ministry of higher education and research and the Tunisian Ministry of higher education and scientific research in the CMCU project number 13G1501 ” Graphes, Géométrie et théorie Spectrale”. Also I would like to thank the Laboratory of Mathematics Jean Leray of Nantes (LMJL) and the research unity (UR/13ES47) of Faculty of Sciences of Bizerte (University of Carthage) for its financial and its continuous support.