Vipul: Created page with "==Definition== Suppose $G$ is a finite undirected graph. Let $n$ be the size of the vertex set $V(G)$ . Fix a bijective c..."

2014-05-25T21:30:08Z

Created page with "==Definition== Suppose <math>G</math> is a finite undirected graph. Let <math>n</math> be the size of the vertex set <math>V(G)</math>. Fix a bijective c..."

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==Definition==

Suppose <math>G</math> is a [[finite graph|finite]] [[undirected graph]]. Let <math>n</math> be the size of the vertex set <math>V(G)</math>. Fix a bijective correspondence <math>v:\{ 1,2,\dots,n\} \to V(G)</math>. The '''normalized Laplacian matrix''', also called the '''symmetric normalized Laplacian matrix''', of <math>G</math>, corresponding to this choice of bijection <math>v</math>, is a <math>n \times n</math> [[linear:square matrix|square matrix]] denoted <math>\mathcal{L}</math> and can be defined in the following equivalent ways:

* It is the matrix <math>\mathcal{L} = D^{-1/2}LD^{-1/2}</math> where <math>D</math> is the [[defining ingredient::degree matrix]] and <math>L</math> is the [[defining ingredient::Laplacian matrix]] of <math>G</matH> (both matrices are written using the vertex ordering given by <math>v</math>).
* It is the matrix <math>\mathcal{L} = I - D^{-1/2}AD^{-1/2}</math> where <math>I</math> is the <math>n \times n</math> [[linear:identity matrix|identity matrix]], <math>D</math> is the [[defining ingredient::degree matrix]], and <math>A</math> is the [[defining ingredient::adjacency matrix]] of <math>G</math> (both matrices are written using the vertex ordering given by <math>v</math>).

Normalized Laplacian matrix - Revision history

Vipul: Created page with "==Definition== Suppose G is a finite undirected graph. Let n be the size of the vertex set V(G). Fix a bijective c..."

Vipul: Created page with "==Definition== Suppose $G$ is a finite undirected graph. Let $n$ be the size of the vertex set $V(G)$ . Fix a bijective c..."