Laplacian Eigenvectors Of Graphs
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Laplacian Eigenvectors Of Graphs
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Author : Türker Biyikoglu
language : en
Publisher: Springer
Release Date : 2007-07-07
Laplacian Eigenvectors Of Graphs written by Türker Biyikoglu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-07-07 with Mathematics categories.
This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.
Laplacian Eigenvectors Of Graphs
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Author : Türker Biyikoglu
language : en
Publisher: Springer
Release Date : 2007-07-26
Laplacian Eigenvectors Of Graphs written by Türker Biyikoglu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-07-26 with Mathematics categories.
This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.
Locating Eigenvalues In Graphs
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Author : Carlos Hoppen
language : en
Publisher: Springer Nature
Release Date : 2022-09-21
Locating Eigenvalues In Graphs written by Carlos Hoppen and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-21 with Mathematics categories.
This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own. Studies in spectral graph theory seek to determine properties of a graph through matrices associated with it. It turns out that eigenvalues and eigenvectors have surprisingly many connections with the structure of a graph. This book approaches this subject under the perspective of eigenvalue location algorithms. These are algorithms that, given a symmetric graph matrix M and a real interval I, return the number of eigenvalues of M that lie in I. Since the algorithms described here are typically very fast, they allow one to quickly approximate the value of any eigenvalue, which is a basic step in most applications of spectral graph theory. Moreover, these algorithms are convenient theoretical tools for proving bounds on eigenvalues and their multiplicities, which was quite useful to solve longstanding open problems in the area. This book brings these algorithms together, revealing how similar they are in spirit, and presents some of their main applications. This work can be of special interest to graduate students and researchers in spectral graph theory, and to any mathematician who wishes to know more about eigenvalues associated with graphs. It can also serve as a compact textbook for short courses on the topic.
Inequalities For Graph Eigenvalues
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Author : Zoran Stanić
language : en
Publisher: Cambridge University Press
Release Date : 2015-07-23
Inequalities For Graph Eigenvalues written by Zoran Stanić and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-07-23 with Mathematics categories.
This book explores the inequalities for eigenvalues of the six matrices associated with graphs. Includes the main results and selected applications.
Distribution Of Laplacian Eigenvalues Of Graphs
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Author : Bilal Ahmad Rather
language : en
Publisher: A.K. Publications
Release Date : 2022-12-22
Distribution Of Laplacian Eigenvalues Of Graphs written by Bilal Ahmad Rather and has been published by A.K. Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-22 with Mathematics categories.
Spectral graph theory (Algebraic graph theory) is the study of spectral properties of matrices associated to graphs. The spectral properties include the study of characteristic polynomial, eigenvalues and eigenvectors of matrices associated to graphs. This also includes the graphs associated to algebraic structures like groups, rings and vector spaces. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in mathematical chemistry (theoretical/quantum chemistry). One of the major problems in spectral graph theory lies in finding the spectrum of matrices associated to graphs completely or in terms of spectrum of simpler matrices associated with the structure of the graph. Another problem which is worth to mention is to characterise the extremal graphs among all the graphs or among a special class of graphs with respect to a given graph, like spectral radius, the second largest eigenvalue, the smallest eigenvalue, the second smallest eigenvalue, the graph energy and multiplicities of the eigenvalues that can be associated with the graph matrix. The main aim is to discuss the principal properties and structure of a graph from its eigenvalues. It has been observed that the eigenvalues of graphs are closely related to all graph parameters, linking one property to another. Spectral graph theory has a wide range of applications to other areas of mathematical science and to other areas of sciences which include Computer Science, Physics, Chemistry, Biology, Statistics, Engineering etc. The study of graph eigen- values has rich connections with many other areas of mathematics. An important development is the interaction between spectral graph theory and differential geometry. There is an interesting connection between spectral Riemannian geometry and spectral graph theory. Graph operations help in partitioning of the embedding space, maximising inter-cluster affinity and minimising inter-cluster proximity. Spectral graph theory plays a major role in deforming the embedding spaces in geometry. Graph spectra helps us in making conclusions that we cannot recognize the shapes of solids by their sounds. Algebraic spectral methods are also useful in studying the groups and the rings in a new light. This new developing field investigates the spectrum of graphs associated with the algebraic structures like groups and rings. The main motive to study these algebraic structures graphically using spectral analysis is to explore several properties of interest.
Graph Embeddings And Laplacian Eigenvalues
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Author : Stephen Guattery
language : en
Publisher:
Release Date : 1998
Graph Embeddings And Laplacian Eigenvalues written by Stephen Guattery and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Embeddings (Mathematics) categories.
Abstract: "Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n x n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix [gamma]; the best possible bound based on this embedding is n/[lambda][subscript max]([gamma superscript T gamma]). However, the best bounds produced by embedding techniques are not tight; they can be off by a factor proportional to log2n for some Laplacians. We show that this gap is a result of the representation of the embedding: by including edge directions in the embedding matrix representation [gamma], it is possible to find an embedding such that [gamma superscript T gamma] has eigenvalues that can be put into a one-to-one correspondence with the eigenvalues of the Laplacian. Specifically, if [lambda] is a nonzero eigenvalue of either matrix, then n/[lambda] is an eigenvalue of the other. Simple transformations map the corresponding eigenvectors to each other. The embedding that produces these correspondences has a simple description in electrical terms if the underlying graph of the Laplaciain [sic] is viewed as a resistive circuit. We also show that a similar technique works for star embeddings when the Laplacian has a zero Dirichlet boundary condition, though the related eigenvalues in this case are reciprocals of each other. In the Dirichlet boundary case, the embedding matrix [gamma] can be used to construct the inverse of the Laplacian. Finally, we connect our results with previous techniques for producing bounds, and provide an illustrative example."
Eigenvectors Of Graphs
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Author :
language : en
Publisher:
Release Date : 1988
Eigenvectors Of Graphs written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with categories.
This grant has supported work in several areas. 1) A study of graph eigenvectors shows connections to graph structure in ways that are reminiscent of eigenfunctions of the laplacian operator in two or three dimensions. Methods developed in this study have also led to estimates of the maximum possible value for the kth eigenvalue of a graph as function of the number of edges or vertices. 2) The convex hull of the rows of an eigenmatrix of a graph is the polytope of an eigenvalue. We investigated relations between such polytopes and the graph. The graph of such a polytope may be isomorphic to the original graph this is the case for most regular polytopes. For distance-regular graphs and several kinds of less symmetric graphs, we can show that the polytope of some eigenvalue has the same group of automorphisms as the graph, that proximity of points is equivalent to adjacency of vertices, and that other properties of the polytope carry over the graph. Possible directions for future work include the following. Determine the reducibility of the group of automorphisms of a polytope and the significance in the graph of faces and facets of the polytopes. Investigate the intrinsic eigenvectors of a graph (the list of inner products of vertices of a polytope with the normal to a supporting hyperplane is an intrinsic eigenvector). Seek physical models for the interpretation of graph eigenvalues and eigenvectors, e.g., transient temperature distributions in a graph-like collection of heat-conducting rods.
Learning Sparse Graph Laplacian With K Eigenvector Prior Via Iterative Glasso And Projection
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Author : Saghar Bagheri
language : en
Publisher:
Release Date : 2021
Learning Sparse Graph Laplacian With K Eigenvector Prior Via Iterative Glasso And Projection written by Saghar Bagheri and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.
Learning a suitable graph is an important precursor to many graph signal processing (GSP) tasks, such as graph signal compression and denoising. Previous graph learning algorithms either make assumptions on graph connectivity (e.g., graph sparsity), or make individual edge weight assumptions such as positive edges only. In this thesis, given an empirical covariance matrix computed from data as input, an eigen-structural assumption on the graph Laplacian matrix is considered: the first K eigenvectors of the graph Laplacian are pre-selected, e.g., based on domain-specific criteria, and the remaining eigenvectors are then learned from data. One example use case is image coding, where the first eigenvector is pre-chosen to be constant, regardless of available observed data. Experimental results show that given the first K eigenvectors as a prior, the algorithm in this thesis outperforms competing graph learning schemes using a variety of graph comparison metrics.
Spectral Graph Theory
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Author : Fan R. K. Chung
language : en
Publisher: American Mathematical Soc.
Release Date :
Spectral Graph Theory written by Fan R. K. Chung and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
Beautifully written and elegantly presented, this book is based on 10 lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University. Chung's well-written exposition can be likened to a conversation with a good teacher - one who not only gives you the facts, but tells you what is really going on, why it is worth doing, and how it is related to familiar ideas in other areas. The monograph is accessible to the nonexpert who is interested in reading about this evolving area of mathematics.
Partitioning Sparse Matrices With Eigenvectors Of Graphs
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Author : Alex Pothen
language : en
Publisher:
Release Date : 1989
Partitioning Sparse Matrices With Eigenvectors Of Graphs written by Alex Pothen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Sparse matrices categories.
Abstract: "The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach to computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvalues can be used to compute good separators in grid graphs.
