1. Laplacian矩阵是对称半正定矩阵；
2. Laplacian矩阵的最小特征值是，相应的特征向量是；
3. Laplacian矩阵有个非负实特征值：，且对于任何一个实向量，都有下面的式子成立：

1. 对于给定的图，求图的度矩阵和邻接矩阵；
2. 计算图的Laplacian矩阵；
3. 对Laplacian矩阵进行特征值分解，取其前个特征值对应的特征向量，构成的特征向量矩阵；
4. 利用K-Means聚类算法对上述的的特征向量矩阵进行聚类，每一行代表一个样本点。

2、利用相似度矩阵的构造方法

#coding:UTF-8
'''
Created on 2015年5月12日

@author: zhaozhiyong
'''
from __future__ import division
import scipy.io as scio
from scipy import sparse
from scipy.sparse.linalg.eigen import arpack#这里只能这么做，不然始终找不到函数eigs
from numpy import *


def spectalCluster(data, sigma, num_clusters):
    print "将邻接矩阵转换成相似矩阵"
    #先完成sigma ！= 0
    print "Fixed-sigma谱聚类"
    data = sparse.csc_matrix.multiply(data, data)

    data = -data / (2 * sigma * sigma)
    
    S = sparse.csc_matrix.expm1(data) + sparse.csc_matrix.multiply(sparse.csc_matrix.sign(data), sparse.csc_matrix.sign(data))   
    
    #转换成Laplacian矩阵
    print "将相似矩阵转换成Laplacian矩阵"
    D = S.sum(1)#相似矩阵是对称矩阵
    D = sqrt(1 / D)
    n = len(D)
    D = D.T
    D = sparse.spdiags(D, 0, n, n)
    L = D * S * D
    
    #求特征值和特征向量
    print "求特征值和特征向量"
    vals, vecs = arpack.eigs(L, k=num_clusters,tol=0,which="LM")  
    
    # 利用k-Means
    print "利用K-Means对特征向量聚类"
    #对vecs做正规化
    sq_sum = sqrt(multiply(vecs,vecs).sum(1))
    m_1, m_2 = shape(vecs)
    for i in xrange(m_1):
        for j in xrange(m_2):
            vecs[i,j] = vecs[i,j]/sq_sum[i]
    
    myCentroids, clustAssing = kMeans(vecs, num_clusters)
    
    for i in xrange(shape(clustAssing)[0]):
        print clustAssing[i,0]
    

def randCent(dataSet, k):
    n = shape(dataSet)[1]
    centroids = mat(zeros((k,n)))#create centroid mat
    for j in range(n):#create random cluster centers, within bounds of each dimension
        minJ = min(dataSet[:,j]) 
        rangeJ = float(max(dataSet[:,j]) - minJ)
        centroids[:,j] = mat(minJ + rangeJ * random.rand(k,1))
    return centroids

def distEclud(vecA, vecB):
    return sqrt(sum(power(vecA - vecB, 2))) #la.norm(vecA-vecB)

def kMeans(dataSet, k):
    m = shape(dataSet)[0]
    clusterAssment = mat(zeros((m,2)))#create mat to assign data points to a centroid, also holds SE of each point
    centroids = randCent(dataSet, k)
    clusterChanged = True
    while clusterChanged:
        clusterChanged = False
        for i in range(m):#for each data point assign it to the closest centroid
            minDist = inf; minIndex = -1
            for j in range(k):
                distJI = distEclud(centroids[j,:],dataSet[i,:])
                if distJI < minDist:
                    minDist = distJI; minIndex = j
            if clusterAssment[i,0] != minIndex: clusterChanged = True
            clusterAssment[i,:] = minIndex,minDist**2
        #print centroids
        for cent in range(k):#recalculate centroids
            ptsInClust = dataSet[nonzero(clusterAssment[:,0].A==cent)[0]]#get all the point in this cluster
            centroids[cent,:] = mean(ptsInClust, axis=0) #assign centroid to mean 
    return centroids, clusterAssment


if __name__ == '__main__':
    # 导入数据集
    matf = 'E://data_sc//corel_50_NN_sym_distance.mat'
    dataDic = scio.loadmat(matf)
    data = dataDic['A']
    # 谱聚类的过程
    spectalCluster(data, 20, 18)

2、网上提供的一个Matlab代码

function [cluster_labels evd_time kmeans_time total_time] = sc(A, sigma, num_clusters)
%SC Spectral clustering using a sparse similarity matrix (t-nearest-neighbor).
%
%   Input  : A              : N-by-N sparse distance matrix, where
%                             N is the number of data
%            sigma          : sigma value used in computing similarity,
%                             if 0, apply self-tunning technique
%            num_clusters   : number of clusters
%
%   Output : cluster_labels : N-by-1 vector containing cluster labels
%            evd_time       : running time for eigendecomposition
%            kmeans_time    : running time for k-means
%            total_time     : total running time

%
% Convert the sparse distance matrix to a sparse similarity matrix,
% where S = exp^(-(A^2 / 2*sigma^2)).
% Note: This step can be ignored if A is sparse similarity matrix.
%
disp('Converting distance matrix to similarity matrix...');
tic;
n = size(A, 1);

if (sigma == 0) % Selftuning spectral clustering
  % Find the count of nonzero for each column
  disp('Selftuning spectral clustering...');
  col_count = sum(A~=0, 1)';
  col_sum = sum(A, 1)';
  col_mean = col_sum ./ col_count;
  [x y val] = find(A);
  A = sparse(x, y, -val.*val./col_mean(x)./col_mean(y)./2);
  clear col_count col_sum col_mean x y val;
else % Fixed-sigma spectral clustering
  disp('Fixed-sigma spectral clustering...');
  A = A.*A;
  A = -A/(2*sigma*sigma);
end

% Do exp function sequentially because of memory limitation
num = 2000;
num_iter = ceil(n/num);
S = sparse([]);
for i = 1:num_iter
  start_index = 1 + (i-1)*num;
  end_index = min(i*num, n);
  S1 = spfun(@exp, A(:,start_index:end_index)); % sparse exponential func
  S = [S S1];
  clear S1;
end
clear A;
toc;

%
% Do laplacian, L = D^(-1/2) * S * D^(-1/2)
%
disp('Doing Laplacian...');
D = sum(S, 2) + (1e-10);
D = sqrt(1./D); % D^(-1/2)
D = spdiags(D, 0, n, n);
L = D * S * D;
clear D S;
time1 = toc;

%
% Do eigendecomposition, if L =
%   D^(-1/2) * S * D(-1/2)    : set 'LM' (Largest Magnitude), or
%   I - D^(-1/2) * S * D(-1/2): set 'SM' (Smallest Magnitude).
%
disp('Performing eigendecomposition...');
OPTS.disp = 0;
[V, val] = eigs(L, num_clusters, 'LM', OPTS);
time2 = toc;

%
% Do k-means
%
disp('Performing kmeans...');
% Normalize each row to be of unit length
sq_sum = sqrt(sum(V.*V, 2)) + 1e-20;
U = V ./ repmat(sq_sum, 1, num_clusters);
clear sq_sum V;
cluster_labels = k_means(U, [], num_clusters);
total_time = toc;

%
% Calculate and show time statistics
%
evd_time = time2 - time1
kmeans_time = total_time - time2
total_time
disp('Finished!');

function cluster_labels = k_means(data, centers, num_clusters)
%K_MEANS Euclidean k-means clustering algorithm.
%
%   Input    : data           : N-by-D data matrix, where N is the number of data,
%                               D is the number of dimensions
%              centers        : K-by-D matrix, where K is num_clusters, or
%                               'random', random initialization, or
%                               [], empty matrix, orthogonal initialization
%              num_clusters   : Number of clusters
%
%   Output   : cluster_labels : N-by-1 vector of cluster assignment
%
%   Reference: Dimitrios Zeimpekis, Efstratios Gallopoulos, 2006.
%              http://scgroup.hpclab.ceid.upatras.gr/scgroup/Projects/TMG/

%
% Parameter setting
%
iter = 0;
qold = inf;
threshold = 0.001;

%
% Check if with initial centers
%
if strcmp(centers, 'random')
  disp('Random initialization...');
  centers = random_init(data, num_clusters);
elseif isempty(centers)
  disp('Orthogonal initialization...');
  centers = orth_init(data, num_clusters);
end

%
% Double type is required for sparse matrix multiply
%
data = double(data);
centers = double(centers);

%
% Calculate the distance (square) between data and centers
%
n = size(data, 1);
x = sum(data.*data, 2)';
X = x(ones(num_clusters, 1), :);
y = sum(centers.*centers, 2);
Y = y(:, ones(n, 1));
P = X + Y - 2*centers*data';

%
% Main program
%
while 1
  iter = iter + 1;

  % Find the closest cluster for each data point
  [val, ind] = min(P, [], 1);
  % Sum up data points within each cluster
  P = sparse(ind, 1:n, 1, num_clusters, n);
  centers = P*data;
  % Size of each cluster, for cluster whose size is 0 we keep it empty
  cluster_size = P*ones(n, 1);
  % For empty clusters, initialize again
  zero_cluster = find(cluster_size==0);
  if length(zero_cluster) > 0
    disp('Zero centroid. Initialize again...');
    centers(zero_cluster, :)= random_init(data, length(zero_cluster));
    cluster_size(zero_cluster) = 1;
  end
  % Update centers
  centers = spdiags(1./cluster_size, 0, num_clusters, num_clusters)*centers;

  % Update distance (square) to new centers
  y = sum(centers.*centers, 2);
  Y = y(:, ones(n, 1));
  P = X + Y - 2*centers*data';

  % Calculate objective function value
  qnew = sum(sum(sparse(ind, 1:n, 1, size(P, 1), size(P, 2)).*P));
  mesg = sprintf('Iteration %d:\n\tQold=%g\t\tQnew=%g', iter, full(qold), full(qnew));
  disp(mesg);

  % Check if objective function value is less than/equal to threshold
  if threshold >= abs((qnew-qold)/qold)
    mesg = sprintf('\nkmeans converged!');
    disp(mesg);
    break;
  end
  qold = qnew;
end

cluster_labels = ind';


%-----------------------------------------------------------------------------
function init_centers = random_init(data, num_clusters)
%RANDOM_INIT Initialize centroids choosing num_clusters rows of data at random
%
%   Input : data         : N-by-D data matrix, where N is the number of data,
%                          D is the number of dimensions
%           num_clusters : Number of clusters
%
%   Output: init_centers : K-by-D matrix, where K is num_clusters
rand('twister', sum(100*clock));
init_centers = data(ceil(size(data, 1)*rand(1, num_clusters)), :);

function init_centers = orth_init(data, num_clusters)
%ORTH_INIT Initialize orthogonal centers for k-means clustering algorithm.
%
%   Input : data         : N-by-D data matrix, where N is the number of data,
%                          D is the number of dimensions
%           num_clusters : Number of clusters
%
%   Output: init_centers : K-by-D matrix, where K is num_clusters

%
% Find the num_clusters centers which are orthogonal to each other
%
Uniq = unique(data, 'rows'); % Avoid duplicate centers
num = size(Uniq, 1);
first = ceil(rand(1)*num); % Randomly select the first center
init_centers = zeros(num_clusters, size(data, 2)); % Storage for centers
init_centers(1, :) = Uniq(first, :);
Uniq(first, :) = [];
c = zeros(num-1, 1); % Accumalated orthogonal values to existing centers for non-centers
% Find the rest num_clusters-1 centers
for j = 2:num_clusters
  c = c + abs(Uniq*init_centers(j-1, :)');
  [minimum, i] = min(c); % Select the most orthogonal one as next center
  init_centers(j, :) = Uniq(i, :);
  Uniq(i, :) = [];
  c(i) = [];
end
clear c Uniq;