拉普拉斯矩阵(Laplacian Matrix) 及半正定性证明

摘自 https://blog.csdn.net/beiyangdashu/article/details/49300479

和 https://en.wikipedia.org/wiki/Laplacian_matrix

定义

给定一个由n个顶点的简单图G，它的拉普拉斯矩阵 $L_{{n imes n}}$

$L = D - A，$ $其中，D是该图G度的矩阵，A为图G的邻接矩阵。$

$因为G是一个简单图，A只包含0,1，并且它的对角元素均为0.$

$L中的元素给定为：$

$L_{{i,j}}:={egin{cases}deg(v_{i})&{mbox{if}} i=j\-1&{mbox{if}} i eq j {mbox{and}} v_{i}{mbox{ is adjacent to }}v_{j}\0&{mbox{otherwise}}end{cases}}$

$其中deg(v i) 表示顶点 i 的度。$

$对称归一化的拉普拉斯 (Symmetric normalized Laplacian)$

$对称归一化的拉普拉斯矩阵定义为：$

L^{{{ ext{sym}}}}:=D^{{-1/2}}LD^{{-1/2}}=I-D^{{-1/2}}AD^{{-1/2}}

$L^{{{ ext{sym}}}}$

$L_{{i,j}}^{{{ ext{sym}}}}:={egin{cases}1&{mbox{if}} i=j {mbox{and}} deg(v_{i}) eq 0\-{frac {1}{{sqrt {deg(v_{i})deg(v_{j})}}}}&{mbox{if}} i eq j {mbox{and}} v_{i}{mbox{ is adjacent to }}v_{j}\0&{mbox{otherwise}}.end{cases}}$

随机游走归一化的拉普拉斯 (Random walk normalized Laplacian)

随机游走归一化的拉普拉斯矩阵定义为：

L^{{{ ext{rw}}}}:=D^{{-1}}L=I-D^{{-1}}A

$L^{{{ ext{rw}}}}$

$L_{{i,j}}^{{{ ext{rw}}}}:={egin{cases}1&{mbox{if}} i=j {mbox{and}} deg(v_{i}) eq 0\-{frac {1}{deg(v_{i})}}&{mbox{if}} i eq j {mbox{and}} v_{i}{mbox{ is adjacent to }}v_{j}\0&{mbox{otherwise}}.end{cases}}$

泛化的拉普拉斯 (Generalized Laplacian)

泛化的拉普拉斯Q定义为:

${displaystyle {egin{cases}Q_{i,j}<0&{mbox{if}} i eq j {mbox{and}} v_{i}{mbox{ is adjacent to }}v_{j}\Q_{i,j}=0&{mbox{if}} i eq j {mbox{and}} v_{i}{mbox{ is not adjacent to }}v_{j}\{mbox{any number}}&{mbox{otherwise}}.end{cases}}}$

注意：普通的拉普拉斯矩阵为泛化的拉普拉斯矩阵。

例子

Labeled graph	Degree matrix	Adjacency matrix	Laplacian matrix
	$left(egin{array}{rrrrrr} 2 & 0 & 0 & 0 & 0 & 0\ 0 & 3 & 0 & 0 & 0 & 0\ 0 & 0 & 2 & 0 & 0 & 0\ 0 & 0 & 0 & 3 & 0 & 0\ 0 & 0 & 0 & 0 & 3 & 0\ 0 & 0 & 0 & 0 & 0 & 1\ end{array} ight)$	$left(egin{array}{rrrrrr} 0 & 1 & 0 & 0 & 1 & 0\ 1 & 0 & 1 & 0 & 1 & 0\ 0 & 1 & 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0 & 1 & 1\ 1 & 1 & 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1 & 0 & 0\ end{array} ight)$	$left(egin{array}{rrrrrr} 2 & -1 & 0 & 0 & -1 & 0\ -1 & 3 & -1 & 0 & -1 & 0\ 0 & -1 & 2 & -1 & 0 & 0\ 0 & 0 & -1 & 3 & -1 & -1\ -1 & -1 & 0 & -1 & 3 & 0\ 0 & 0 & 0 & -1 & 0 & 1\ end{array} ight)$