graph Laplacian
E577498
The graph Laplacian is a matrix representation of a graph that encodes its connectivity and is fundamental in spectral graph theory, clustering, and network analysis.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
graph-theoretic concept
ⓘ
matrix ⓘ operator ⓘ |
| alsoKnownAs | Laplacian matrix NERFINISHED ⓘ |
| captures |
combinatorial structure of a graph
ⓘ
connectivity properties of a graph ⓘ |
| centralConceptIn | spectral graph theory NERFINISHED ⓘ |
| definedOn | vertices of a graph ⓘ |
| dependsOn |
adjacency matrix of the graph
ⓘ
degree matrix of the graph ⓘ |
| eigenvaluesCalled | graph spectrum ⓘ |
| encodes | graph connectivity ⓘ |
| field |
graph theory
ⓘ
network science ⓘ spectral graph theory ⓘ |
| hasEigenvalue | 0 ⓘ |
| hasVariant |
combinatorial Laplacian
ⓘ
normalized Laplacian ⓘ random-walk Laplacian ⓘ signless Laplacian ⓘ |
| isPositiveSemidefinite | true ⓘ |
| isSymmetric | true for undirected graphs ⓘ |
| kernelDimensionEquals | number of connected components of the graph ⓘ |
| matrixSize | n-by-n for a graph with n vertices ⓘ |
| relatedTo |
continuous Laplace operator
ⓘ
discrete Laplace operator NERFINISHED ⓘ |
| secondSmallestEigenvalueName | algebraic connectivity ⓘ |
| secondSmallestEigenvalueSymbol | Fiedler value NERFINISHED ⓘ |
| secondSmallestEigenvectorName | Fiedler vector NERFINISHED ⓘ |
| smallestEigenvalue | 0 ⓘ |
| usedFor |
Cheeger inequality applications
ⓘ
community detection ⓘ diffusion processes on graphs ⓘ dimensionality reduction on graphs ⓘ graph partitioning ⓘ graph signal processing ⓘ manifold learning ⓘ network robustness analysis ⓘ random walk analysis ⓘ semi-supervised learning on graphs ⓘ spectral clustering ⓘ spectral embedding ⓘ |
| usedIn |
Markov chains
NERFINISHED
ⓘ
data clustering ⓘ electrical network theory ⓘ image segmentation ⓘ machine learning ⓘ network analysis ⓘ numerical analysis ⓘ physics of networks ⓘ |
Referenced by (1)
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