Laplacian Matrix - Definition

Definition

Given a simple graph G with n vertices, its Laplacian matrix is defined as:

$L = D - A.$

That is, it is the difference of the degree matrix D and the adjacency matrix A of the graph. In the case of directed graphs, either the indegree or outdegree might be used, depending on the application.

From the definition follows that:

$\ell_{i,j}:= \begin{cases} \deg(v_i) & \mbox{if}\ i = j \\ -1 & \mbox{if}\ i \neq j\ \mbox{and}\ v_i \mbox{ is adjacent to } v_j \\ 0 & \mbox{otherwise} \end{cases}$

where deg(v_i) is degree of the vertex i.

The normalized Laplacian matrix is defined as:

$\ell_{i,j}:= \begin{cases} 1 & \mbox{if}\ i = j\ \mbox{and}\ \deg(v_i) \neq 0\\ -\frac{1}{\sqrt{\deg(v_i)\deg(v_j)}} & \mbox{if}\ i \neq j\ \mbox{and}\ v_i \mbox{ is adjacent to } v_j \\ 0 & \mbox{otherwise}. \end{cases}$

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