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Spectral graph theory
About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.
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TL;DR: In this paper, the Laplacian matrix of a simple graph G = (V, E) is defined as L(G) = D (G) - A(G).
Abstract: Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertexdegrees and by A(G) its adjacency matrix. Then, the Laplacian matrix of G is L(G) = D(G) - A(G). The first and second section of this paper contains introduction and some known results, respectively. The third section is devoted to properties of Laplacian spectrum. The fourth section contains characterization of graphs. The fifth section relates the Laplacian eigenvalues with the graph structure.
126 citations
19 May 2012
TL;DR: In this paper, it was shown that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O(√(λk log k).
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. It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into Rk, and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n/k and λk, the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O(√(λk log k)). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result [LRTV12]. The √(log k) bound is tight, up to constant factors, for the "noisy hypercube" graphs.
124 citations
TL;DR: An iterative graph-based framework for image restoration based on a new definition of the normalized graph Laplacian, which comprises of outer and inner iterations, where in each outer iteration, the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations.
Abstract: Any image can be represented as a function defined on a weighted graph, in which the underlying structure of the image is encoded in kernel similarity and associated Laplacian matrices. In this paper, we develop an iterative graph-based framework for image restoration based on a new definition of the normalized graph Laplacian. We propose a cost function, which consists of a new data fidelity term and regularization term derived from the specific definition of the normalized graph Laplacian. The normalizing coefficients used in the definition of the Laplacian and associated regularization term are obtained using fast symmetry preserving matrix balancing. This results in some desired spectral properties for the normalized Laplacian such as being symmetric, positive semidefinite, and returning zero vector when applied to a constant image. Our algorithm comprises of outer and inner iterations, where in each outer iteration, the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations. This procedure improves the performance of the algorithm for image deblurring, where we do not have access to a good initial estimate of the underlying image. In addition, the specific form of the cost function allows us to render the spectral analysis for the solutions of the corresponding linear equations. In addition, the proposed approach is general in the sense that we have shown its effectiveness for different restoration problems, including deblurring, denoising, and sharpening. Experimental results verify the effectiveness of the proposed algorithm on both synthetic and real examples.
124 citations
TL;DR: In this paper, a spectral theory of graphs based on the signless Laplacian of graphs is proposed. But the spectral theory is restricted to graph angles, and it is not suitable for graph angles.
Abstract: This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results.
123 citations
TL;DR: This paper considers the energy of a simple graph with respect to its normalized Laplacian eigenvalues, which is called the L-energy, and provides upper and lower bounds for L- energy based on its general Randic index R-1(G).
122 citations