Showing papers on "Spectral graph theory published in 2000"
TL;DR: This work treats image segmentation as a graph partitioning problem and proposes a novel global criterion, the normalized cut, for segmenting the graph, which measures both the total dissimilarity between the different groups as well as the total similarity within the groups.
Abstract: We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.
13,789 citations
TL;DR: In this article, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second degree of the graph.
Abstract: In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.
68 citations
TL;DR: In this article, the spectral properties of the operator (- + V (x )) on a graph were studied and an expression for the spectral determinant was derived, which generalizes one obtained for the Laplacian operator.
Abstract: We study the spectral properties of the operator (- +V (x )) on a graph. ( is the Laplacian and V (x ) is some potential defined on the graph). In particular, we derive an expression for the spectral determinant that generalizes one previously obtained for the Laplacian operator.
40 citations
TL;DR: In this paper, an efficient method is developed for decomposition of finite element meshes based on concepts from algebraic graph theory and consists of an efficient algorithm to calculate the Fiedler vector of the Laplacian matrix of a graph.
Abstract: In this paper an efficient method is developed for decomposition of finite element meshes. The present method is based on concepts from algebraic graph theory and consists of an efficient algorithm to calculate the Fiedler vector of the Laplacian matrix of a graph. The problem of finding the second eigenvalue of the Laplacian matrix is converted into that of evaluating the maximal eigenvalue of the complementary Laplacian matrix. The corresponding eigenvector is constructed by a simple iterative method and applied to graph partitioning. An appropriate transformation maps the graph partitioning into that of domain decomposition of the corresponding finite element mesh. Copyright © 2000 John Wiley & Sons, Ltd.
22 citations
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TL;DR: In this paper, it was shown that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. In particular, they derived recurrent relations for the limiting averaged moments of the adjacency matrix of G.
Abstract: We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency matrix of G. These relations allow one to study the corresponding eigenvalue distribution function; we show that its density has an infinite support in contrast to the case of the ordinary discrete Laplacian.
17 citations
TL;DR: In this paper, a scaling limit of the spectral distribution of the Laplacian on the Johnson graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability is discussed.
Abstract: On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function.
11 citations
03 Sep 2000
TL;DR: Here it is shown that the spectral graph analysis method can be used to compute useful correspondences even when the level of point movement is large, and is evaluated on both synthetic and real-world data.
Abstract: This paper investigates the correspondence matching of point-sets using spectral graph analysis. In particular we deal with the problem of how the modal analysis of point-sets can be rendered robust to point movement. To this end we consider alternatives to the Gaussian proximity matrix. We evaluate the new method on both synthetic and real-world data. Here we show that the method can be used to compute useful correspondences even when the level of point movement is large.
5 citations
01 Oct 2000
5 citations
TL;DR: How simple computations on the eigenvalues of the laplacian matrix of a graph and a certain Gram matrix provide bounds on the number of edges joining parts in a graph where the vertex set is separated in parts of given sizes is described.