Topic
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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01 Jan 2006
TL;DR: In this paper, the concept of matrix-valued ef-fective resistances for undirected matrix-weighted graphs is introduced, which are defined to be the square blocks that appear in the diagonal of the inverse of the matrix weighted Dirichlet graph Laplacian matrix.
Abstract: We introduce the concept of matrix-valued ef- fective resistance for undirected matrix-weighted graphs. Ef- fective resistances are defined to be the square blocks that appear in the diagonal of the inverse of the matrix-weighted Dirichlet graph Laplacian matrix. However, they can also be obtained from a "generalized" electrical network that is constructed from the graph, and for which currents, voltages and resistances take matrix values. Effective resistances play an important role in several problems related to distributed control and estimation. They appear in least-squares estimation problems in which one attempts to reconstruct global information from relative noisy measurements; as well as in motion control problems in which agents attempt to control their positions towards a desired formation, based on noisy local measurements. We show that in either of these problems, the effective resistances have a direct physical interpretation. We also show that effective resistances provide bounds on the spectrum of the graph Laplacian matrix and the Dirichlet graph Laplacian. These bounds can be used to characterize the stability and convergence rate of several distributed algorithms that appeared in the literature.
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TL;DR: In this paper , the Laplacian spectrum of the directed windmill graph with k ≥ 1 was determined, and the characteristic polynomial of the matrix of the graph was determined.
Abstract: Suppose that 0 = µ0 ≤ µ1 ≤ ... ≤ µn-1 are eigen values of a Laplacian matrix graph with n vertices and m(µ0), m(µ1), …, m(µn-1) are the multiplicity of each µ, so the Laplacian spectrum of a graph can be expressed as a matrix 2 × n whose line elements are µ0, µ1, …, µn-1 for the first row, and m(µ0), m(µ1), …, m(µn-1) for the second row. In this paper, we will discuss Laplacian spectrum of the directed windmill graph () with k ≥ 1. The determination of the Laplacian spectrum in this study is to determine the characteristic polynomial of the Laplacian matrix from the directed windmill graph () with k ≥ 1.
Keywords: Characteristic polynomial, directed windmill graph, Laplacian matrix, Laplacian spectrum.MSC2020 :05C50
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05 Jan 2023TL;DR: In this article , the spectral clustering of graph Laplacians has been studied, which can be unnormalized or normalized, and a regularized version of the adjacency matrix is discussed.
Abstract: The previous two chapters have focused on the problem of graph partitioning, which has seen enormous interest and research work in recent years. We continue that aspect of network analysis by introducing the notion of spectral clustering. The main tool of this chapter is the graph Laplacian, which can be unnormalized or normalized. Also discussed is a regularized version of the adjacency matrix.
01 Jan 2005
TL;DR: An efficient analytical method is presented for calculating the eigenvalues of special matrices related to finite element meshes (FEMs) with regular topologies using a skeleton graph as the model of a FEM.
Abstract: In this paper an efficient analytical method is presented for calculating the eigenvalues of special matrices related to finite element meshes (FEMs) with regular topologies. In the proposed method, a skeleton graph is used as the model of a FEM. This graph is then considered as the Cartesian product of its generators. The eigenvalues of the Laplacian matrix of the entire graph are then easily calculated using the eigenvalues of its generators. An exceptionally fast method is also proposed for computing the second eigenvalue of the Laplacian of the graph model of a FEM, known as the Fiedler vector. After ordering the entries of the second eigenvector, the graph model is partitioned and the corresponding FEM is bisected.
01 Jun 2012
TL;DR: Various features such as brightness features, shape features and statistical features are stated and Bayes classifier using Gaussian mixture model is used as classifier and feature extraction method based on spectral graph theory is presented.
Abstract: For exact classification of the defect, good feature selection and classifier is necessary. In this paper, various features such as brightness features, shape features and statistical features are stated and Bayes classifier using Gaussian mixture model is used as classifier. Also feature extraction method based on spectral graph theory is presented. Experimental result shows that feature extraction method using graph Laplacian result in better performance than the result using PCA.