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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.


Papers
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Journal ArticleDOI
19 Jun 2012
TL;DR: In this paper, Fourier analytic proofs are given for several results recently obtained by Solymosi ((33)), Vu ((41)) and Vinh ((40)) using spectral graph theory.
Abstract: In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi ((33)), Vu ((41)) and Vinh ((40)) using spectral graph theory. In addition, several gener- alizations of these results are given. In the case that A is a subset of a prime field Fp of size less than p 1/2 it is shown that |{a 2 + b : a, b ∈ A}| ≥ C|A| 147/146 , where | � | denotes the cardinality of the set and C is an absolute constant.

31 citations

Proceedings ArticleDOI
06 Jan 2019
TL;DR: In this paper, the authors introduced the notion of a submodular transformation F : {0, 1}n → Rm, which applies m sub-modular functions to the n-dimensional input vector.
Abstract: The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations.In this paper, we introduce the notion of a submodular transformation F : {0, 1}n → Rm, which applies m submodular functions to the n-dimensional input vector, and then introduce the notions of its Laplacian and normalized Laplacian. With these notions, we unify and generalize the existing Cheeger inequalities by showing a Cheeger inequality for submodular transformations, which relates the conductance of a submodular transformation and the smallest non-trivial eigenvalue of its normalized Laplacian. This result recovers the Cheeger inequalities for undirected graphs, directed graphs, and hypergraphs, and derives novel Cheeger inequalities for mutual information and directed information.Computing the smallest non-trivial eigenvalue of a normalized Laplacian of a submodular transformation is NP-hard under the small set expansion hypothesis. In this paper, we present a polynomial-time O(log n)-approximation algorithm for the symmetric case, which is tight, and a polynomial-time O(log2n + log n · log m)-approximation algorithm for the general case.We expect the algebra concerned with submodular transformations, or submodular algebra, to be useful in the future not only for generalizing spectral graph theory but also for analyzing other problems that involve piecewise linear transformations, e.g., deep learning.

31 citations

Journal ArticleDOI
TL;DR: A new matrix called adjusted adjacency matrix is proposed that meets the requirement that a graph must contain at least one distinct eigenvalue and is shown to be not only effective but also more efficient than that based on the adjACency matrix.
Abstract: Many science and engineering problems can be represented by a network, a generalization of which is a graph. Examples of the problems that can be represented by a graph include: cyclic sequential circuit, organic molecule structures, mechanical structures, etc. The most fundamental issue with these problems (e.g., designing a molecule structure) is the identification of structure, which further reduces to be the identification of graph. The problem of the identification of graph is called graph isomorphism. The graph isomorphism problem is an NP problem according to the computational complexity theory. Numerous methods and algorithms have been proposed to solve this problem. Elsewhere we presented an approach called the eigensystem approach. This approach is based on a combination of eigenvalue and eigenvector which are further associated with the adjacency matrix. The eigensystem approach has been shown to be very effective but requires that a graph must contain at least one distinct eigenvalue. The adjacency matrix is not shown sufficiently to meet this requirement. In this paper, we propose a new matrix called adjusted adjacency matrix that meets this requirement. We show that the eigensystem approach based on the adjusted adjacency matrix is not only effective but also more efficient than that based on the adjacency matrix.

31 citations

Book
01 Jan 2017
TL;DR: In this article, the adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph.
Abstract: Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterises certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.

30 citations

Proceedings ArticleDOI
05 Jun 2016
TL;DR: The spectral graph sparsi-fication algorithm can efficiently build an ultra-sparse subgraph from a spanning tree subgraph by adding a few “spectrally-critical” off-tree edges back to the spanning tree, enabled by a novel spectral perturbation approach and allows to approximately preserve key spectral properties of the original graph Laplacian.
Abstract: Spectral graph sparsification aims to find an ultra-sparse subgraph whose Laplacian matrix can well approximate the original Laplacian matrix in terms of its eigenvalues and eigenvectors. The resultant sparsified subgraph can be efficiently leveraged as a proxy in a variety of numerical computation applications and graph-based algorithms. This paper introduces a practically efficient, nearly-linear time spectral graph sparsification algorithm that can immediately lead to the development of nearly-linear time symmetric diagonally-dominant (SDD) matrix solvers. Our spectral graph sparsi-fication algorithm can efficiently build an ultra-sparse subgraph from a spanning tree subgraph by adding a few “spectrally-critical” off-tree edges back to the spanning tree, which is enabled by a novel spectral perturbation approach and allows to approximately preserve key spectral properties of the original graph Laplacian. Extensive experimental results confirm the nearly-linear runtime scalability of an SDD matrix solver for large-scale, real-world problems, such as VLSI, thermal and finite-element analysis problems, etc. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within two minutes using a single CPU core and about 6GB memory.

30 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981