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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 ...
It is used together with the adjacency matrix to construct the Laplacian matrix of a graph: ... the degree matrix for is a diagonal matrix defined as [1],:= ...
As a consequence, the spherical Laplacian of a function defined on S N−1 ⊂ R N can be computed as the ordinary Laplacian of the function extended to R N ∖{0} so that it is constant along rays, i.e., homogeneous of degree zero.
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.
Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph, which is equal to the difference between the graph's degree matrix (the diagonal matrix of vertex degrees) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise).
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.
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace.It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to ...
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.