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2012•Unpublished venueRequires access

Multi-way spectral partitioning and higher-order cheeger inequalities

James R. Lee, Shayan Oveis Gharan, Luca Trevisan

Open publisher page 118 citations

Abstract

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.

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What this paper is about

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.

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OpenAlex reports 118 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.

Key concepts: Spectral graph theory, Laplacian matrix, Algebraic connectivity, Eigenvalues and eigenvectors, Mathematics, Integral graph, Graph energy, Combinatorics

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