Laplace matrix (Laplacian matrix) and semi-positive qualitative proof

Excerpt from 49300479

and Https://en.wikipedia.org/wiki/Laplacian_matrix

Defined

Given a simple graph g of n vertices, its Laplace matrix is defined as:

L = d-a, where D is the matrix of the G degree of the graph, and A is the adjacency matrix of Figure G.

Because G is a simple graph, a contains only 0, 1, and its diagonal elements are all 0.

The elements in L are given as:

where deg (VI) represents the degree of vertex i.

symmetric normalized laplace (symmetric normalized Laplacian)

The Laplace matrix of symmetric normalization is defined as:

,

The element given is:

Random Walk normalized Laplace (random walk normalized Laplacian)

The Laplace matrix of a random walk normalization is defined as:

is given as an element of

Generalized Laplace (generalized Laplacian)

The generalized Laplace q definition is:

Note: The ordinary Laplace matrix is a generalized Laplace matrix.

Example

labeled Graph	degree Matrix	adjacency Matrix	Laplacian Matrix

Semi-positive qualitative proof of Laplace matrix

Laplace matrix (Laplacian matrix) and semi-positive qualitative proof