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Showing papers on "Spectral graph theory published in 2001"


Journal ArticleDOI
TL;DR: In this paper, the extremal graphs for which equality in de Caen's inequality holds and then apply the inequality to give an upper bound for the largest Laplacian eigenvalue λ 1 (G) of a graph.

68 citations


Journal ArticleDOI
TL;DR: In this article, the authors generalize a result in (R. Merris, Port. Math. 48 (3) 1991) and obtain the following result: Let G be a graph and M(G) be a maximum matching in G.

51 citations


Journal ArticleDOI
TL;DR: An exact analysis of two conductor-insulator transitions in the random graph model where low connectivity means high impurity concentration is presented and the height of the delta peak at zero energy in its spectrum is computed exactly.
Abstract: We present an exact analysis of two conductor-insulator transitions in the random graph model where low connectivity means high impurity concentration. The adjacency matrix of the random graph is used as a hopping Hamiltonian. We compute the height of the delta peak at zero energy in its spectrum exactly and describe analytically the structure and contribution of localized eigenvectors. The system is a conductor for average connectivities between 1.421 529ellipsis and 3.154 985ellipsis but an insulator in the other regimes. We explain the spectral singularity at average connectivity e = 2.718 281ellipsis and relate it to other enumerative problems in random graph theory.

34 citations


DOI
01 Jan 2001
TL;DR: The nor- malized Laplacian spectrum of AS graphs is unique in spite of the explosive growth of the Internet and distinctive in setting AS graphs apart from synthetic ones, suggesting it is an excellent candidate as a concise finger print of Internet-like graphs.
Abstract: In this paper we investigate properties of the Internet topology on the AS (autonomous system) level. Among techniques in spectral graph theory, we find the nor- malized Laplacian spectrum ( ) of AS graphs 1) unique in spite of the explosive growth of the Internet and 2) distinctive in setting AS graphs apart from synthetic ones. These prop- erties suggest that is an excellent candidate as a concise finger print of Internet-like graphs. Further analysis into the theory of leads us to a new structural classification of AS graphs with plausible inter- pretations in networking terms. Extensive analysis by AS- level data supports this claim. More importantly, along the way, new power-law relationships are unveiled, giving rise to a hybrid model encompassing both structural and power- law properties. We think that these new insights may have a profound impact on future protocol evaluation and design.

34 citations


Book ChapterDOI
01 Jan 2001
TL;DR: The Laplacian is another important matrix associated with a graph, and the spectrum is the spectrum of this matrix as mentioned in this paper, and it can be used to provide interesting geometric representations of a graph.
Abstract: The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

17 citations


Journal ArticleDOI
TL;DR: In this paper, a weighted incidence graph is constructed for the finite element model and a spectral partitioning heuristic is applied to the graph using the second and the third eigenvalues of the Laplacian matrix of the graph, to partition it into three subgraphs and correspondingly trisect the finite elements model.
Abstract: In this paper a new method is proposed for finite element domain decomposition. A weighted incidence graph is first constructed for the finite element model. A spectral partitioning heuristic is then applied to the graph using the second and the third eigenvalues of the Laplacian matrix of the graph, to partition it into three subgraphs and correspondingly trisect the finite element model.

14 citations


Proceedings ArticleDOI
01 Jan 2001
TL;DR: A spectral method for graph-matching in which the graph adjacency matrix is taken to represent the transition probability matrix of a Markov chain by developing a brushfire search method to assign correspondences between nodes using the rank-order of the eigenvector co-efficients in first-order n eighbourhoods of the graphs.
Abstract: This paper describes a spectral method for graph-matching. We adopt a graphical models viewpoint in which the graph adjacency matrix is taken to represent the transition probability matrix of a Markov chain. The nodeorder of the steady state random walk associated with this Markov chain is determined by the co-efficent order of the leading eigenvector of the adjacency matrix. We match nodes in different graphs by aligning their sequence order in the steady-state walk. The method proceeds from the nodes with the largest leading eigenvector co-efficient. We develop a brushfire search method to assign correspondences between nodes using the rank-order of the eigenvector co-efficients in first-order n eighbourhoods of the graphs. We demonstrate the utility of the new graph-matching method on both synthetic and real graphs.

13 citations


Journal ArticleDOI
TL;DR: This analysis is then used for obtaining estimates on the spectral condition number of some weighted graph matrices, obtaining lower and upper estimates that are asymptotically tight.
Abstract: We study the extreme singular values of incidence graph matrices, obtaining lower and upper estimates that are asymptotically tight. This analysis is then used for obtaining estimates on the spectral condition number of some weighted graph matrices. A short discussion on possible preconditioning strategies within interior-point methods for network flow problems is also included.

11 citations


Proceedings ArticleDOI
26 Sep 2001
TL;DR: A novel algorithm for shape matching based on a relationship descriptor called the shape context is devised, which enables us to compute similarity measures between shapes which, together with similarity measures for texture and color, can be used for object recognition.
Abstract: We develop a two-stage framework for parsing and understanding images, a process of image segmentation grouping pixels to form regions of coherent color and texture, and a process of recognition - comparing assemblies of such regions, hypothesized to correspond to a single object, with views of stored prototypes. We treat segmenting images into regions as an optimization problem: partition the image into regions such that there is high similarity within a region and low similarity across regions. This is formalized as the minimization of the normalized cut between regions. Using ideas from spectral graph theory, the minimization can be set as an eigenvalue problem. Visual attributes such as color, texture, contour and motion are encoded in this framework by suitable specification of graph edge weights. The recognition problem requires us to compare assemblies of image regions with previously stored proto-typical views of known objects. We have devised a novel algorithm for shape matching based on a relationship descriptor called the shape context. This enables us to compute similarity measures between shapes which, together with similarity measures for texture and color, can be used for object recognition. The shape matching algorithm has yielded excellent results on a variety of different 2D and 3D recognition problems.

10 citations


Dissertation
20 Nov 2001

3 citations


Journal Article
TL;DR: In this paper, the authors considered the problem of finding families of graphs with polylogarithmic spectrum in the number of vertices, where the spectral number of distinct eigenvalues of the adjacency matrix is known.
Abstract: One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Especially load balancing can be done on such interconnection topologies in a small number of steps. In this paper we are interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigenvalues. Our goal is to find scalable families of such graphs with polylogarithmic spectrum in the number of vertices. We present also the eigenvalues of the Butterfly graph.

Journal ArticleDOI
TL;DR: Using a technique developed by A. Nilli (1991, Discrete Math), from above the Cheeger number of a finite connected graph G of small degree (?(G)?5) admitting sufficiently distant edges is estimated.

Book ChapterDOI
15 Feb 2001
TL;DR: This paper is interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigen values, and the eigenvalues of the Butterfly graph are presented.
Abstract: One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Especially load balancing can be done on such interconnection topologies in a small number of steps. In this paper we are interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigenvalues. Our goal is to find scalable families of such graphs with polylogarithmic spectrum in the number of vertices. We present also the eigenvalues of the Butterfly graph.