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In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method.

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rdfs:label	Laplacian matrix (en)
rdfs:comment	In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. (en)
sameAs	Laplacian matrix Laplacian matrix
dbp:wikiPageUsesTemplate	dbt:ISBN dbt:Reflist dbt:Short_description dbt:Use_American_English dbt:Matrix_classes
Subject	Algebraic graph theory Matrices Numerical differential equations
gold:hypernym	Representation
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Laplacian_matrix?oldid=1074004261&ns=0
Wikipage page ID	1448472 (xsd:integer)
page length (characters) of wiki page	39550 (xsd:nonNegativeInteger)
Wikipage revision ID	1074004261 (xsd:integer)
Link from a Wikipage to another Wikipage	Machine learning Nodal admittance matrix Nonlinear dimensionality reduction OR gate Probability vector Adjacency matrix Algebraic connectivity Block matrix Stencil (numerical analysis) Béla Bollobás Calculus on finite weighted graphs Algebraic graph theory Glossary of graph theory Matrices Numerical differential equations Directed graph Discrete Laplace operator Discrete Poisson equation Distance matrix Pierre-Simon Laplace Cheeger constant (graph theory) Cut (graph theory) Definite matrix Degree (graph theory) Degree matrix Kirchhoff's theorem Component (graph theory) Eigendecomposition of a matrix Eigenvalues and eigenvectors Spanning tree Spectral gap Spectral graph theory Spectral layout Finite difference method Graph Graph Fourier transform Graph drawing Graph labeling Graph theory Incidence matrix Kernel (linear algebra) Symmetric matrix Spectral clustering Spring system Diagonally dominant matrix Transition rate matrix Transpose Standard basis Stiffness matrix Stochastic matrix Laplace operator M-matrix Mathematics Matrix (mathematics) Matrix similarity Classical mechanics Neumann boundary condition Random walk Regular graph Resistance distance Signal processing
has abstract	In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph. Together with Kirchhoff's theorem, it can be used to calculate the number of spanning trees for a given graph. The sparsest cut of a graph can be approximated through the second smallest eigenvalue of its Laplacian by Cheeger's inequality. The spectral decomposition of the Laplacian matrix allows constructing low dimensional embeddings that appear in many machine learning applications and determines a spectral layout in graph drawing. The Laplacian matrix is the easiest to define for a simple graph, but more common in applications for a edge-weighted graph, i.e., with weights on its edges — the entries of the graph adjacency matrix. Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries — resulting in normalized adjacency and Laplacian matrixes. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Laplacian_matrix
is Wikipage redirect of	Kirchhoff matrix Kirchhoff matrix (of a graph) Graph Laplacian Laplace matrix Laplacian matrix of a graph
is Link from a Wikipage to another Wikipage of	List of things named after Pierre-Simon Laplace Abelian sandpile model Adjacency matrix Algebraic connectivity Algebraic graph theory Anisotropic Network Model Nullity (graph theory) Braunstein–Ghosh–Severini entropy Brouwer's conjecture Calculus on finite weighted graphs Diffusion map Discrete Laplace operator

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