graph laplacian matrix

Graph Laplacian matrix is a matrix that represents a graph, and it is used in various applications such as machine learning, computer vision, and network analysis. The graph Laplacian matrix is defined as the difference between the degree matrix and the adjacency matrix of the graph.

The degree matrix is a diagonal matrix that contains the degree of each vertex in the graph, while the adjacency matrix is a matrix that represents the connections between vertices.

The graph Laplacian matrix has several properties that make it useful in various applications. For example, it is positive semi-definite, meaning that all of its eigenvalues are non-negative. This property makes it useful in spectral clustering, where the eigenvalues and eigenvectors of the graph Laplacian matrix are used to partition the graph into clusters.

There are several types of graph Laplacian matrices, including the unnormalized Laplacian, the symmetric normalized Laplacian, and the random walk normalized Laplacian. Each of these matrices has its own unique properties and applications.