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A Short Tutorial on Graph Laplacians, Laplacian Embedding ...

A Short Tutorial on Graph Laplacians, LaplacianEmbedding, and Spectral ClusteringRadu HoraudINRIA Grenoble Rhone-Alpes, HoraudGraph Laplacian TutorialIntroductionThespectral Graph theorystudies the properties of graphs viathe eigenvalues and eigenvectors of their associated graphmatrices: theadjacency matrixand thegraph Laplacianandits matrices have been extremely well studied from analgebraic point of Laplacian allows a natural link between discreterepresentations, such as graphs, and continuousrepresentations, such as vector spaces and most important application of the Laplacian isspectralclusteringthat corresponds to a computationally tractablesolution to thegraph partitionning application isspectral matchingthat solves HoraudGraph Laplacian TutorialApplications of spectral Graph theorySpectral partitioning: automatic circuit placement for VLSI(Alpert et al 1999), image segmentation (Shi & Malik 2000),Text mining and web applications.

The spectral graph theory studies the properties of graphs via the eigenvalues and eigenvectors of their associated graph matrices: the adjacency matrix and the graph Laplacian and its variants. Both matrices have been extremely well studied from an algebraic point of view. The Laplacian allows a natural link between discrete

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