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where L is the unnormalized Laplacian, A is the adjacency matrix, D is the degree matrix, and + is the Moore–Penrose inverse. Since the degree matrix D is diagonal, its reciprocal square root ( D + ) 1 / 2 {\textstyle (D^{+})^{1/2}} is just the diagonal matrix whose diagonal entries are the reciprocals of the square roots of the diagonal ...
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex. [1]
The degree matrix indicates the degree of vertices. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph.
Laplacian matrix — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number of spanning trees in the graph. Seidel adjacency matrix — a matrix similar to the usual adjacency matrix but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
The Laplacian is a common operator in image processing and computer vision (see the Laplacian of Gaussian, blob detector, and scale space). The list of formulas in Riemannian geometry contains expressions for the Laplacian in terms of Christoffel symbols. Weyl's lemma (Laplace equation).
The first definition listed for the Laplacian Matrix, L = D-A seems to not match the second definition. With the example graph below, D(1,1) = 4 and A(1,1) = 1 (since there is a loop connecting vertex 1 with itself).