Showing papers on "Spectral graph theory published in 2002"

Spectral kernels for learning machines

Abstract: The spectral kernel machine combines kernel functions and spectral graph theory for solving problems of machine learning The data points in the dataset are placed in the form of a matrix known as a kernel matrix, or Gram matrix, containing all pairwise kernels between the data points The dataset is regarded as nodes of a fully connected graph A weight equal to the kernel between the two nodes is assigned to each edge of the graph The adjacency matrix of the graph is equivalent to the kernel matrix, also known as the Gram matrix The eigenvectors and their corresponding eigenvalues provide information about the properties of the graph, and thus, the dataset The second eigenvector can be thresholded to approximate the class assignment of graph nodes Eigenvectors of the kernel matrix may be used to assign unlabeled data to clusters, merge information from labeled and unlabeled data by transduction, provide model selection information for other kernels, detect novelties or anomalies and/or clean data, and perform supervised learning tasks such as classification

On the Laplacian Eigenvalues and Metric Parameters of Hypergraphs

Abstract: The aim of this paper is to generalize some concepts and recent results of the algebraic graph theory in order to investigate and describe, by algebraic methods, the properties of some combinatorial structures. Here we introduce a version of "Laplacian matrix" of a hypergraph and we obtain several spectral-like results on its metric parameters, such as the diameter, mean distance, excess, bandwidth and cutsets.

On Spectral Integral Variations of Graphs

Abstract: Let G be a general graph. The spectrum S ( G ) of G is defined to be the spectrum of its Laplacian matrix. Let G + e be the graph obtained from G by adding an edge or a loop e . We study in this paper when the spectral variation between G and G + e is integral and obtain some equivalent conditions, through which a new Laplacian integral graph can be constructed from a known Laplacian integral graph by adding an edge.

On Cheeger-type inequalities for weighted graphs

Abstract: We give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger-type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger-type bound available. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 1–17, 2002

Diffusions on graphs, Poisson problems and spectral geometry

Abstract: We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function.

Alignment using Spectral Clusters

Abstract: This paper describes a hierarchical spectral method for the correspondence matching of point-sets. Conventional spectral methods for correspondence matching are notoriously susceptible to differences in the relational structure of the point-sets under consideration. In this paper we demonstrate how the method can be rendered robust to structural differences by adopting a hierarchical approach. We show how the point-clusters associated with the most significant spectral modes can be used to locate correspondences when significant contamination is present. Spectral graph theory is a term applied to a family of techniques that aim to characterise the global structural properties of graphs using the eigenvalues and eigenvectors of the adjacency matrix [1]. Although the subject has found widespread use in a number of areas including structural chemistry and routeing theory, there have been relatively few applications in the computer vision literature. The reason for this is that although elegant, spectral graph representations are notoriously susceptible to the effect of structural error. In other words, spectral graph theory can furnish very efficient methods for characterising exact relational structures, but soon breaks down when there are spurious nodes and edges in the graphs under study. There are several concrete examples in the pattern analysis literature. Umeyama has an eigendecomposition method that recovers the permutation matrix that maximises the correlation or overlap of the adjacency matrices for graphs of the same size [13]. Horaud and Sossa [5] have adopted a purely structural approach to the recognition of linedrawings. Their representation is based on the immanantal polynomials for the Laplacian matrix of the line-connectivity graph. By comparing the coefficients of the polynomials, they are able to index into a large data-base of line-drawings. Shapiro and Brady [11] have developed a method which draws on a representation which uses weighted edges. They commence from a weighted adjacency matrix (or proximity matrix) which is obtained using a Gaussian function of the distances between pairs of points. The eigen-vectors of the adjacency matrix can be viewed as the basis vectors of an orthogonal transformation on the original point identities. In other words, the components of the eigenvectors represent mixing angles for the transformed points. Matching between different point-sets is effected by comparing the pattern of eigenvectors in different images. Finally, a number of authors have used spectral methods to perform pairwise clustering on image data. Shi and Malik [12] use the second eigenvalue to segment grey-scale images by performing an eigen-decomposition on a matrix of pairwise attribute differences. Inoue and Urahama [6] have shown how the sequential extraction of eigen-modes can be used to cluster pairwise

The independent and principal component of graph spectra

Abstract: In this paper, we demonstrate how PCA and ICA can be used for embedding graphs in pattern-spaces. Graph spectral feature vectors are calculated from the leading eigenvalues and eigenvectors of the unweighted graph adjacency matrix. The vectors are then embedded in a lower dimensional pattern space using both the PCA and ICA decomposition methods. Synthetic and real sequences are tested using the proposed graph clustering algorithm. The preliminary results show that generally speaking the ICA is better than PCA for clustering graphs. The best choice of graph spectral feature for clustering is the cluster shared perimeters.

Cheeger Constant and Connectivity of Graphs

Abstract: In the branches of both differential geometry and graph theory, Cheeger constant plays a central role in the study of eigenvalues of Laplacians. In this paper, we give a new aspect of Cheeger constant of graphs, that is, some relations of Cheeger constant and connectivities of digraphs and graphs.

Object recognition by clustering spectral features

Abstract: We investigate whether vectors of graph spectral features can be used for the purposes of graph clustering. We commence from the eigenvalues and eigenvectors of the adjacency matrix. Each of the leading eigenmodes represents a cluster of nodes and is mapped to a component of a feature vector. The spectral features used as components of the vectors are the eigenvalues and the shared perimeter length. We explore whether these vectors can be used for the purposes of graph clustering. Here we investigate the use of both central and pairwise clustering methods. On a database of view-graphs, both of the features provide good clusters while the eigenvectors perform better.

Some properties of characteristic polynomial in graph

Abstract: The characteristic polynomial of graph have many pr operties. In this paper the exponent expression of characteristic polyno mial are gived, several different expression of derivative of characteristi c polynomial and the graph theory meaning of derivatives of higher order are als o gived.

Graph-Theoretical Method for Rouse-Ham Dynamics.

Abstract: A new graph-theoretical method for calculating the dynamical properties of Rouse-Ham chains with any branches is presented. The characteristic polynomial for its line graph, which has the bonds of a molecular graph as its beads with adjacency of bonds as in the graph, makes it possible to provide us with the relaxation spectrum of any tree-like chain.

On spectra of expansion graphs and

Abstract: An expansion graph of a directed weighted graph G0 is obtained from G0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum up to a fixed matrix, are compared. A limiting formula is proved for the sequence of the spectral radii of a sequence of expansion graphs of G0 when the lengths of all chains replacing some original edges tend to infinity. It is shown that for all expansion graphs of G0 the adjacency matrices have the same level characteristic, but they can have different height characteristics as examples show.