\documentclass[12pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb}
\usepackage{euscript,graphicx,subfigure,times,version}
\usepackage{pstricks}
\include{amsmath_def}
\include{symbols}
\excludeversion{hide}
\parindent=0in


\def\pS {\partial S}
\def\Sb {\overline{S}}
\def\hPG {h_{P}(G)}
\def\hAG {h_{A}(G)}
\def\mi {m_{\infty}}
\def\DA {\varDelta_{A}}
\def\DP {\varDelta_{P}}
\def\pG {\partial G}
\def\pV {\partial V}
\def\pE {\partial E}
\def\Vz {\stackrel{o}{V}}
\def\Ez {\stackrel{o}{E}}


\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 20: Introduction to Spectral Graph Theory--III. Graph Cut
and Cheeger Constants, Isospectral Graphs, and Discrete Laplacian
Eigenvalue Problems}
\author{Lecturer: Naoki Saito\\
        Scribe: Brendan Farrell/Allen Xue}
\date{June 6, 2007}
\begin{document}
\maketitle


\begin{section}{The Cheeger Ratio and the Cheeger Constants of a Graph}
This lecture is based primarily on material in \cite{Ura02} and
\cite[Sec. 2.1-2.3]{Chu97}.

\begin{defn}
Given a graph $G=(V,E)$. Let $S\subset V$ be a subset of vertices.
Then $\pS\define \{e=(x,y)\in E\cond x\in S,\ y\notin S\}$,
\end{defn}

\begin{ex}
In Figure~\ref{fig:1}, $S$ is the set of dark nodes, and the dotted
lines form $\pS$. $\Sb\define V\backslash S$. Also,
$\text{vol}(S)=m(S)\define \sum_{x\in S}m(x)$.
\end{ex}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm, height=3.5cm]{lecture20-n1.eps}
\caption{$S$ consists of the dark nodes. The dotted lines form
$\pS$.} \label{fig:1}
\end{center}
\end{figure}


\begin{defn}
The {\bf Cheeger ratio} for $S\subset V$ is defined as
\[h(S)\define \fr{\#(\pS)}{\min(\text{vol}(S),\text{vol}(\Sb))}=\fr{|\pS|}{\min(\text{vol}(S),\text{vol}(\Sb))},\]
\end{defn}

\begin{ex}
In Figure~\ref{fig:1}, $\text{vol}(S)=4+5+4+5=18$ and
$\text{vol}(\Sb)=3+2+3+2+2=12$. $\#(\pS)=8$. So the Cheeger ratio is
$h(S)=8/12=2/3$.
\end{ex}

The Cheeger ratio tells about the quality of the {\bf cut} of $V$
into $S\cup \Sb=V$. In Figure~\ref{fig:2}, a graph that nearly
separates into two separate graphs is shown. Note that this graph is
well balanced, i.e., $\text{vol}(S)\approx \text{vol}(\Sb)$; and
there exist few connections between $S$ and $\Sb$, i.e., $|\pS|$ is
small. In such case, this cut will give us small $h(S)$.

\begin{figure}[!h]
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(0,0)(8,1)
\uput[90](1,0.1){$S$} \uput[90](7,0.1){$\overline{S}$}
\end{pspicture}
\includegraphics[width=8cm, height=3.5cm]{lecture20-n2.eps}
\caption{The cut into $S$ and $\Sb$ generates a small Cheeger
ratio.} \label{fig:2}
\end{center}
\end{figure}

\begin{defn}
The {\bf Cheeger constants} $\hPG$ and $\hAG$ are defined as
\[\hPG\define \inf_{S\subset V,\ S\neq\emptyset} h(S),\]
\[\hAG\define \inf_{S\subset V,\ S\neq\emptyset} \fr{\#(\pS)}{\min\{\#(S),\#(\overline{S})\}}=\inf_{S\subset V,\ S\neq\emptyset}\fr{|\pS|}{\min(|S|,|\Sb|)}.\]
\end{defn}

%This is used for image segmentation.
%Each pixel of a photo is assigned a vector of measurement, such as
%the 3d vector for r-g-b coloring. Then the matrix of the distances
%between these vectors is used to create a weighted graph. The graph
%is then cut into sections corresponding to distinct regions of the
%image. Applications of spectral graph theory to image segmentation
%and clustering have been a hot topic since the late 1990s. The is a
%huge number of paper written in the area.  A few of the basic papers
%in the area are \cite{SM00,Wei99,GS06,BN03,KVV04}. The papers
%finally lead to the differential geometry of Raphy Coifmann's group
%that are summarized in a special issue of the journal Applied and
%Computational Harmonic Analysis \cite{DC06}.


The following theorem provides the upper bounds for $\hPG$ and
$\hAG$.

\begin{thm}~\\
\begin{enumerate}
\item Dodzink and Kendall (1986) \cite{DK86}: $\hPG \leq \sqrt{2\lambda_{2}^{(P)}}$, i.e., $\lambda_{2}^{(P)}\geq
\fr{h^{2}_{P}(G)}{2}$;
\item Mohar (1987) \cite{Moh89}: $\hAG\lneqq\sqrt{ \lambda_{2}^{(A)}(2\mi(G)-\lambda_{2}^{(A)})}\quad$ for $p=\#(V)\geq
4$;
\item Tan (2003) \cite{Tan03}: $\hPG\leq\sqrt{ \lambda_{2}^{(P)}(2-\lambda_{2}^{(P)})}\quad $, for $p\geq 4$. Equality holds if and only if $\ G=K_{1,p-1}$.
\end{enumerate}
\end{thm}
These inequalities are used to evaluate the lower bounds of
$\lambda_2^{(P)}$, $\lambda_2^{(A)}$.

\ \\
For the upper bounds of the eigenvalues, we have
\begin{thm}[Urakawa 1999 \cite{Ura99}]~\\

For $2\leq j\leq \lfloor \frac{diam(G)}{2}\rfloor$,

\[ \lambda_{j}^{(A)}\leq \mi(G)-2\sqrt{\mi(G)-1}\cos\left(\frac{\pi}{\frac{diam(G)}{2j}+1} \right);\]

\[ \lambda_{j}^{(P)}\leq 1-\frac{2\sqrt{\mi(G)-1}}{\mi(G)}\cos\left(\frac{\pi}{\frac{diam(G)}{2j}+1} \right).\]

\end{thm}

Note that there also exists the simpler upper bound:
\[ \lambda_{2}^{(P)}\leq 2 \hPG.\]

Let us prove this inequality.

\begin{proof}
Using the minimum principle ($\mathrm{MP}_{2}$), we have
\begin{eqnarray*}
\lambda_{2}^{(P)}&=&\inf_{f\in C_{0}(V),\ \inner{f}{1}=0,\ f\not\equiv
0}\fr{\inner{\dd{f}}{\dd{f}}_0}{\|f\|^2}\\[8pt]
&=&\inf_{f\in C_{0}(V),\ \inner{f}{1}=0,\ f\not\equiv 0}
\fr{\inner{\DA
f}{\dd{f}}_0}{\|f\|^{2}}\\[8pt]
&=&\inf_{f\in C_{0}(V),\ \inner{f}{1}=0,\ f\not\equiv 0}\fr{
\Sum{x\sim y}(f(x)-f(y))^{2}}{\Sum{x\in V}m(x)f^{2}(x)},
\end{eqnarray*}

where we have used the following definitions:
\begin{eqnarray*}
\inner{f}{g}_{0}&\define&\Sum{u\in V}f(u)g(u),\ \text{and}\\
\|f\|^{2}&\define&\Sum{u\in V}m(u)f^{2}(u).
\end{eqnarray*}

Now suppose $S$ achieves the Cheeger constant $\hPG$. Set

\[ f(x)\define \left\{ \begin{array}{lcl}
\frac{1}{\text{vol}(S)}&\text{if}&x\in S,\\[6pt]
-\frac{1}{\text{vol}(\Sb)}&\text{if}&x\notin S.
\end{array}\right. \]

Then
\begin{eqnarray*}
\inner{f}{1} & = & \Sum{x\in V}f(x)\cdot 1\cdot m(x)\\
&=&\Sum{x\in S}\fr{1}{\text{vol}(S)}m(x) +\Sum{x\in \Sb}\(-\fr{1}{\text{vol}(\Sb)}\)m(x)\\
&=& \fr{\text{vol}(S)}{\text{vol}(S)}-\fr{\text{vol}(\Sb)}{\text{vol}(\Sb)}\\
&=&0.
\end{eqnarray*}

We also have
\begin{eqnarray*}
\Vert f\Vert^2 & = & \Sum{x\in V}f^{2}(x)m(x)\\
&=& \Sum{x\in S}\fr{m(x)}{\text{vol}(S)^{2}} +\Sum{x\in \Sb}\fr{m(x)}{\text{vol}(\Sb)^{2}}\\
&=& \fr{1}{\text{vol}(S)}+\fr{1}{\text{vol}(\Sb)}.
\end{eqnarray*}

Moreover,
\begin{eqnarray*}
\inner{\dd{f}}{\dd{f}}_{0} & = & \inner{\DA f}{f}_{0}\\
&=&\Sum{x\sim y} (f(x)-f(y))^{2}\\
&=&\Sum{{x\sim y \atop x,y\in S}}(f(x)-f(y))^{2}+\Sum{{x\sim y \atop x\in S,y\in \Sb}}(f(x)-f(y))^{2}+\Sum{{x\sim y \atop x,y\in \Sb}}(f(x)-f(y))^{2}\\
&=&0+|\pS|\( \fr{1}{\text{vol}(S)}+\fr{1}{\text{vol}(\Sb)}\)^{2}+0.
\end{eqnarray*}

Now, using the inequality $\frac{a+b}{ab}\leq \frac{2}{\min{(a,b)}}$
for $a,b> 0$, we find
\begin{eqnarray*}
\lambda_{2}^{(P)}&\leq & \fr{|\pS|\cdot
\(\fr{1}{\text{vol}(S)}+\fr{1}{\text{vol}(\Sb)}\)^{2}}{\fr{1}{\text{vol}(S)}+\fr{1}{\text{vol}(\Sb)}}\\[6pt]
&=&|\pS|\(\fr{1}{\text{vol}(S)}+\fr{1}{\text{vol}(\Sb)}\)\\[6pt]
&\leq & \fr{2|\pS|}{\min(\text{vol}(S),\text{vol}(\Sb))}\\[6pt]
&=&2\hPG.
\end{eqnarray*}
The last equality is derived by the assumption that $S$ achieves
$\hPG$.

\end{proof}

\end{section}

\begin{section}{Isospectrality}

\begin{defn}
Let $G=(V,E)$. Define $\text{Spec}(G,A)$, $\text{Spec}(G,\DA)$ and
$\text{Spec}(G,\DP)$ as the sets of spectra (eigenvalues) of $A$,
$\DA$ and $\DP$ of $G$, respectively. For given $G_{1}$ and $G_{2}$,
if $\text{Spec}(G_{1},A)=\text{Spec}(G_{2},A)$, then $G_{1}$ and
$G_{2}$ are said to be {\bf cospectral}. If
$\text{Spec}(G_{1},\DA)=\text{Spec}(G_{2},\DA)$ or
$\text{Spec}(G_{1},\DP)=\text{Spec}(G_{2},\DP)$, then $G_{1}$ and
$G_{2}$ are said to be {\bf isospectral}.
\end{defn}

For cospectrality and isospectrality of graphs, see \cite{Fis66},
\cite{Bak66}, \cite{Tan98} and \cite{FK99}. For more on the
isospectral inequalities, see \cite{KU98}, \cite{KU99} and chapter
2. of \cite[Chap. 2]{Chu97}.

\begin{ex} (Fisher 1966 \cite{Fis66}, Baker 1966 \cite{Bak66}).
Figure~\ref{fig:3} shows two cospectral graphs, however they are not
isomorphic.
\end{ex}

\begin{figure}[!h]
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(0,0.5)(12,3.5)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(0.1,2)(1.5,2)
\psline[dotsize=8pt,linewidth=1pt]{-*}(1.5,2)(2.5,3.5)
\psline[dotsize=8pt,linewidth=1pt]{-*}(1.5,2)(2.5,0.5)
\psline[linewidth=1pt]{-}(2.5,3.5)(2.5,0.5)
\psline[dotsize=8pt,linewidth=1pt]{-*}(2.5,3.5)(3.5,2)
\psline[linewidth=1pt]{-}(2.5,0.5)(3.5,2)
\psline[dotsize=8pt,linewidth=1pt]{-*}(3.5,2)(5,2)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(7,3)(11,1)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(7,1)(11,3)
\psline[linewidth=1pt]{-}(7,3)(7,1)
\psline[linewidth=1pt]{-}(11,3)(11,1)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(9,2)(9,3.5)
\end{pspicture}
\caption{Cospectral but not isomorphic graphs} \label{fig:3}
\end{center}
\end{figure}

\begin{ex}(Fuji-Katsuda 1999 \cite{FK99}, Tan 1998 \cite{Tan98}).
Figure~\ref{fig:4} is an example of two $\DA$-isospectral graphs.
\end{ex}

\begin{figure}[!h]
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-1,1)(12,4)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(0.1,3)(2,3)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(0.1,1)(2,1)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(4,3)(4,1)
\psline[linewidth=1pt]{-}(0.1,3)(0.1,1)
\psline[linewidth=1pt]{-}(2,3)(2,1)
\psline[linewidth=1pt]{-}(2,1)(4,3)
\psline[linewidth=1pt]{-}(2,1)(4,1)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(6.5,3)(8.5,3)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(6.5,1)(8.5,1)
\psline[linewidth=1pt]{-}(6.5,3)(6.5,1)
\psline[linewidth=1pt]{-}(8.5,3)(8.5,1)
%\psline[linewidth=1pt]{-}(6.5,1)(8.5,3)
\psline[dotsize=8pt,linewidth=1pt]{-*}(6.5,1)(7.5,2)
\psline[linewidth=1pt]{-}(7.5,2)(8.5,3)
\psline[dotsize=8pt,linewidth=1pt]{-*}(8.5,1)(10.5,1)
\end{pspicture}
\caption{$\DA$-isospectral graphs.} \label{fig:4}
\end{center}
\end{figure}

\begin{ex}(Fuji-Katsuda 1999 \cite{FK99}, Tan 1998 \cite{Tan98}). 
Figure~\ref{fig:5} shows an example of two $\DP$-isospectral graphs.
\end{ex}

\begin{figure}[!h]
\begin{center}
\begin{pspicture}(0,0.5)(12,4)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(4,0.5)(2,3)
\psline[dotsize=8pt,linewidth=1pt]{-*}(4,0.5)(4,3)
\psline[dotsize=8pt,linewidth=1pt]{-*}(4,0.5)(6,3)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(8,3)(10.5,3)
\psline[dotsize=8pt,linewidth=1pt]{*-*}(8,0.5)(10.5,0.5)
\psline[linewidth=1pt]{-}(8,3)(8,0.5)
\psline[linewidth=1pt](10.5,3)(10.5,0.5)
\end{pspicture}
\caption{$\DP$-isospectral graphs.} \label{fig:5}
\end{center}
\end{figure}

\end{section}


\begin{section}{Discrete Laplacian Eigenvalue Problems}
\begin{defn}
$G=(V,E)$ is said to have {\bf boundary} $\pG=(\pV,\pE)$ if the
following two conditions are met:
\begin{enumerate}
\item \[ \left\{ \begin{array}{lr}
V=\Vz\bigcup \pV & \Vz\bigcap \pV=\emptyset;\\
E=\Ez \bigcup \pE &\Ez \bigcap \pE =\emptyset.
\end{array} \right.\]
\item For each $e=(x,y)\in E$, $x,y\in V$,
\begin{eqnarray*}
e\in \Ez & \Longleftrightarrow & x,y\in V;\\
e\in \pE & \Longleftrightarrow & x\in \Vz,\ y\in \pV \text{ or }x\in
\pV,\ y\in \Vz.
\end{eqnarray*}
\end{enumerate}
\end{defn}

\begin{ex}
Note that there exists some arbitrariness of $\pG$. In
Figure~\ref{fig:6}, the open nodes are in $\Vz$, the closed nodes in
$\pV$, the solid lines in $E$, and the dashed lines in $\pE$.
\end{ex}

\begin{figure}[!h]
\begin{center}
\subfigure[]{\includegraphics[width=1.5in]{lecture20-n6.eps}}
\hspace{1.5cm}
\subfigure[]{\includegraphics[width=1.5in]{lecture20-n7.eps}}
\end{center}
\caption{The arbitrariness of $\partial G$.}
\label{fig:6}
\end{figure}



Consider the discrete Laplacian eigenvalue problem, defined as

\[ \left\{ \begin{array}{rcll}
\DP u&=&\mu^{(P)}u& \text{in } \Vz, \\
u&=&0&\text{on }\pV.
\end{array} \text{   Dirichlet-Laplacian $\(\text{D-L}^{(P)}\);$} \right.\]

\[ \left\{ \begin{array}{rcll}
\DP u&=&\nu^{(P)}u& \text{in } \Vz, \\
du&=&0&\text{on }\pE.
\end{array} \text{   Neumann-Laplacian $\(\text{N-L}^{(P)}\).$} \right.\]

Of course, we can define the discrete Laplacian eigenvalue problem
with $\varDelta A$. We can order these eigenvalues as

\[ D-L^{(A)}: \mu_{1}^{(A)}\leq\mu_{2}^{(A)}\leq\ldots\leq \mu_{k}^{(A)},\;\;\;k=\#(\Vz).\]
\[ D-L^{(P)}: \mu_{1}^{(P)}\leq\mu_{2}^{(P)}\leq\ldots\leq \mu_{k}^{(P)},\;\;\;k=\#(\Vz).\]

We have the following theorem.

\begin{thm}
$\mu_{1}^{(P)}>0$, and $\mu_{1}^{(A)}>0$, both with multiplicity
$1$. And there exist $\varphi_{1}^{(P)}(x)>0$ and
$\varphi_{1}^{(A)}(x)>0$ for all $x\in \Vz$.
\end{thm}

\begin{proof}
The proof is essentially the same as the continuum case, i.e., use
the discrete Green's identity with the boundary condition. For
example, let $\varphi^{(P)}_1$ be an eigenfunction for $\mu^{(P)}_1$
with $\Vert\varphi^{(P)}_1\Vert=1$, then using the discrete Green's
identity,

\[\mu_{1}^{(P)} = \inner{\varDelta
\varphi_{1}^{(P)}}{\varphi_{1}^{(P)}} =
\inner{\dd{\varphi^{(P)}_1}}{\dd{\varphi^{(P)}_1}} =
\|\dd{\varphi^{(P)}_1}\|^{2}\geq 0.\]

If $\dd{\varphi^{(P)}_1}\equiv 0$, then
$\varphi^{(P)}_1\equiv\text{const}$ on $V$. But $\varphi^{(P)}_1=0$
on $\pV$ forces us to have $\varphi^{(P)}_1\equiv 0$. So
$\mu^{(P)}_1\neq 0$. Therefore, $\mu_1^{(P)}>0$.

\end{proof}

\begin{thm}[The discrete Faber-Krahn inequality] see \cite{KU98},
\cite{KU99}.~\\

If $\# (\Ez\cup \pE)=n$, then

\[\mu_{1}^{(P)}(L_{n})\leq \mu^{(P)}_{1}(G)\]

where equality holds if and only if $G=L_{n}$. $L_{n}$ a graph is
shown in Figure 7. There are $n$ nodes. Only the last to the right
belongs to $\pV$, the rest belong to $\Vz$, and the same is true for
the edges.
\end{thm}

\begin{figure}[!h]
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(0,0)(8,4)
\psline[dotsize=5pt,linewidth=1pt]{oo-oo}(0.1,1.5)(1.5,1.5)
\uput[270](0.1,1.5){$x_1$} \uput[270](1.5,1.5){$x_2$}
\psline[linewidth=0.6pt]{-}(0.8,1.6)(3,2.5)
\psline[dotsize=5pt,linewidth=1pt]{-oo}(1.5,1.5)(2.8,1.5)
\psline[linewidth=0.6pt]{-}(2.1,1.6)(3,2.5)
\psline[dotsize=5pt,linewidth=1pt]{-oo}(2.8,1.5)(4.2,1.5)
\psline[linewidth=0.6pt]{-}(3.5,1.6)(3,2.5)
\psline[dotsize=5pt,linewidth=1pt,linestyle=dotted]{-oo}(4.2,1.5)(5.2,1.5)
\psline[linewidth=0.6pt]{-}(4.7,1.6)(3,2.5)
\psline[dotsize=5pt,linewidth=1pt]{-oo}(5.2,1.5)(6.2,1.5)
\psline[linewidth=0.6pt]{-}(5.7,1.6)(3,2.5)
\uput[270](6.2,1.5){$x_n$} \uput[270](7.6,1.5){$x_{n+1}$}
\uput[270](7.2,0.6){$\pV$}
\psline[linewidth=0.6pt]{->}(7.4,0.6)(7.6,1)
\psline[dotsize=5pt,linewidth=1pt,linestyle=dashed]{-**}(6.2,1.5)(7.6,1.5)
\psline[linewidth=0.6pt]{<-}(6.8,1.5)(7.4,2) \uput[90](7.4,2){$\pE$}
\uput[0](0,0.6){$\underbrace{~\hspace{5.9cm}}_{\Vz}$}
\uput[90](3,2.6){$\Ez$}
\end{pspicture}
\end{center}
\caption{Graph $L_n$.} \label{fig:Ln}
\end{figure}

%\begin{figure}[h]
%  \begin{center}
%    \includegraphics[width=2in]{pic7.eps}
%  \end{center}
%  \caption{\small $L_{n}$ }%caption. To get a figure to span two
%      columns, use the environment figure* rather than figure.}
 % \label{fig-label}
%\end{figure}
\end{section}

\begin{thebibliography}{10}

\bibitem{Bak66}
{\sc  G. A. Baker}: ``Drum shapes and isospectral graphs'', {\em J.
Math. Phys.}, vol. 7, pp.2238-2242, 1966.

\bibitem{BN03}
{\sc M. Belkin and P. Niyogi}: ``Laplacian eigenmaps for
dimensionality reduction and data representation'', {\em Neural
Comp.}, vol. 15, no. 6, pp. 1373-1396, 2003.

\bibitem{Chu97}
{\sc F. R. K. Chung}: {\em Spectral Graph Theory}, AMS, 1997.

\bibitem{DC06}
{\em Applied and Computational Harmonic Analysis, Special Issue on Diffusion Maps and Wavelets}, (D. Donoho and C. Chui, Eds.), vol. 21, no. 1, pp. 1-144, 2006.

\bibitem{DK86}
{\sc  J. Dodziuk and W. S. Kendall}: ``Combinatorial Laplacians and
isoperimetric inequality'', {\em From Local Times to Global
Geometry, Control and Physics} (K. D. Elworthy, ed.), Pitman Res.
Notes Math. Ser., vol. 150, pp. 68-74, Longman Sci. Tech., 1986.


\bibitem{Fis66}
{\sc M. E. Fisher}: ``On hearing the shape of a drum'', {\em J.
Comb. Theory}, vol. 1, pp. 105-125, 1966.

\bibitem{FK99}
{\sc H. Fujii and A. Katsuda}: ``Isospectral graphs and
isoperimetric constants'', {\em Discrete Math.}, vol. 207, no. 1-3,
pp. 33-52, 1999.

\bibitem{GS06}
{\sc L. Grady and E. L. Schwartz}: ``Isoperimetric graph
partitioning for image segmentation'', {\em IEEE Trans. on Pattern
Anal. Machine Intell.}, vol. 28, no. 3, pp. 469-475, 2006.

\bibitem{KU98}
{\sc  A. Katsuda and H. Urakawa}: ``The first eigenvalue of the
discrete Dirichlet problem for a graph'', {\em J. Combin. Math.
Combin. Comput.} vol. 27, pp. 217-225, 1998.

\bibitem{KU99}
{\sc A. Katsuda and H. Urakawa}: ``The Faber-Krahn type
isoperimetric inequalities for a graph'', {\em Tohoku Math. J.} (2)
vol. 51, no. 2, pp. 267-281, 1999.

\bibitem{KVV04}
{\sc R. Kannan, S. Vempala, and A. Vetta}: ``On clusterings: good,
bad and spectral'', {\em J. Assoc. Comput. Mach.} vol. 51 no. 3, pp.
497-515, 2004.

\bibitem{Moh89}
{\sc B. Mohar}: ``Isoperimetric numbers of graphs'', {\em J. Comb.
Theory}, Ser. B, vol. 47, no. 3, pp. 274-291, 1989.

\bibitem{SM00}
{\sc J. Shi and J. Malik}: ``Normalized cuts and image
segmentation'', {\em IEEE Trans. Pattern Anal. Machine Intell.},
vol. 22, no. 8, pp. 888-905, 2000.


\bibitem{Tan98}
{\sc  J. Tan}: ``On isospectral graphs'', {\em Interdisciplinary
Information Sciences}, vol. 4, no. 2, pp. 117-124, 1998.

\bibitem{Tan03}
{\sc J. Tan}: ``On Cheeger inequalities of a graph'', {\em Discrete
Math}., vol. 269, no. 1-3, pp. 315-323, 2003.

\bibitem{Ura99}
{\sc  H. Urakawa}: ``Eigenvalue comparison theorems of the discrete
Laplacians for a graph'', {\em Geom. Dedicata}, vol. 74 no. 1, pp.
95-112, 1999.

\bibitem{Ura02}
{\sc H. Urakawa}: ``Spectral geometry and graph theory'', {\em Ouyou
Suuri} (Bulletin of the Japan Society of Industrial and Applied
Mathematics), vol. 12, no. 1, pp.29-45, 2002.

\bibitem{Wei99}
{\sc Y. Weiss}: ``Segmentation using eigenvectors: a unifying
view'', in {\em Proc. the Seventh IEEE International Conference on
Computer Vision}, vol. 2, pp. 975-982, 1999.

\end{thebibliography}
\end{document}
