Spectral graph theory chung pdf merge

Secondary sources 1fan rk chung, spectral graph theory, vol. Notes on elementary spectral graph theory applications to. Featured on meta creative commons licensing ui and data updates. It was called the interchange graph by ore 272, derivative by h. Spectral graph theory studies how the eigenvalues of the adjacency matrix of a graph, which are purely algebraic quantities, relate to combinatorial properties of the graph. Important early work was done by social scientists. Fem justification if apply finite element method to solve laplaces equation in the plane with a delaunay triangulation would get graph laplacian, but with some weights on edges fundamental solutions are x and y coordinates see strangs introduction to applied mathematics. Spectral graph theory american mathematical society. Chung s spectral graph theory book focuses mostly on the normalized laplacian, but this is also good to look into. The book was published by ams in 1992 with a second printing in 1997. Spectral algorithms for learning latent variable models by simons institute. Spectral graph theory studies connections between combinatorial properties of graphs and the eigenvalues of matrices associated to the graph, such as the adjacency matrix and the laplacian matrix.

Fem justification if apply finite element method to solve laplaces equation in the plane with a delaunay triangulation would get graph laplacian, but with some weights on edges. Given a graph g, the most obvious matrix to look at is. Cbms regional conference series in mathematics publication year 1997. The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors spectral properties of associated matrices. 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. To construct a spectral sparsifier of an arbitrary unweighted graph, we first apply theorem 3 to find a 1log 2 n spectral decomposition of the graph in which the boundary has at most half the edges. I sometimes edit the notes after class to make them way what i wish i had said. The notes written before class say what i think i should say. Spectral graph theory is the study of properties of the laplacian matrix or adjacency matrix associated with a graph.

Introduction spectral graph theory has a long history. University of pennsylvania, philadelphia, pennsylvania 19104 email address. Notes on elementary spectral graph theory applications to graph clustering using normalized cuts jean gallier department of computer and information science university of pennsylvania philadelphia, pa 19104, usa email. Spectral graph theory revised and improved fan chung. Spielman september 4, 2015 disclaimer these notes are not necessarily an accurate representation of what happened in class.

In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such. Spectral graph theory studies connections between combinatorial properties of graphs and the eigenvalues of matrices associated to the graph, such as the adjacency matrix and the laplacian. Chu fan chung, spectral graph theory, american mathematical society. However, substantial revision is clearly needed as the list of errata got longer. Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random. For instance, star graphs and path graphs are trees. A good book that is available as a free pdf is spectra of graphs by brouwer and haemers i personally am partial to the normalized laplacian in which case you would want to look at spectral graph theory by chung. Spectral graph theory seeks to understand the interesting properties and structure of a graph by using the dominant singular values and vectors, rst recognized by fiedler 1973. Spectral graph theory cbms regional conference series in. In this paper, we focus on the connection between the eigenvalues of the laplacian.

The mathematical life of fan chung american mathematical society. Here we shall concentrate mainly on the adjacency matrix of undirected graphs, and. Spectral graph theory emerged in the 1950s and 1960s. The following theorem is one of the fundamental results in spectral graph theory with many applications in complexity theory, coding theory, analysis of random walks, approximation algorithms, etc. Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Spectral graph theory uri feige january 2010 1 background with every graph or digraph one can associate several di.

Spectral graph theory and its applications lillian dai 6. Notes on elementary spectral graph theory applications to graph clustering using normalized cuts jean gallier department of computer and information science university of pennsylvania philadelphia. In the summer of 2006, the daunting task of revision finally but surely got started. Lectures on spectral graph theory fan rk chung ucsd.

Lecture notes on expansion, sparsest cut, and spectral. Spectral graph theory revised and improved fan chung the book was published by ams in 1992 with a second printing in 1997. Properties and recent applications in spectral graph theory. We then sparsify each of the components by random sampling, and we sparsify the graph formed by the boundary edges recursively. Steps in a proof from spectral graph theory by fan chung.

Techniques from spectral graph theory, linear and multilinear algebra, probability, approximation theory, etc. I personally think biggs is somewhat dated but more accesible for beginners. Spectral properties of hypergraph laplacian and approximation algorithms. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti. Graph analysis provides quantitative tools for the study of complex networks. Based on spectral graph theory chung and graham 1997, spectral. Cbms regional conference series in mathematics, 1997. The fan chung book on spectral graph theory and dan spielmans notes on the same. Chung, 9780821803158, available at book depository with free delivery worldwide. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry, but the connections between these two lines of work were not discovered until much later. Section 2 provides an overview of the spectral graph theory. Spectral graph theory is a discrete analogue of spectral geometry, with the laplacian on a graph being a discrete analogue of the laplacebeltrami operator on a riemannian manifold.

Spectral graph theory is the study and exploration of graphs through the eigenvalues and eigenvectors of matrices naturally associated with those graphs. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The unreasonable effectiveness of spectral graph theory. Laplaces equation and its discrete form, the laplacian matrix, appear ubiquitously in mathematical physics. To merge them further, we need to take the distance between a and b c. Chungs wellwritten exposition can be likened to a conversation with a good teacherone who not only gives you the facts, but tells you what is really going on, why it is.

However, substantial revision is clearly needed as the list of errata got. In the early days, matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Hubert chan anand louis zhihao gavin tang chenzi zhang abstract the celebrated cheegers inequality am85, alo86 establishes a bound on the edge expansion of a graph via its spectrum. We will introduce spectral graph theory by seeing the value of studying the eigenvalues of various. Selected bibliographies on applications of the theory of graph spectra 19 4. This has led to various graphbased neural networks being proposed over the years. Browse other questions tagged graphtheory markovchains spectralgraphtheory or ask your own question. Eigenvalues and the laplacian of a graph chapter 1 2. To construct a spectral sparsifier of an arbitrary unweighted graph, we first apply theorem 3 to find a 1log 2 nspectral decomposition of the graph in which the boundary has at most half the edges. Shapes of uncertainty in spectral graph theory 3 uni. Fiedler number, see godsil and royle 8 chapter and chung 3. Pdf source localization on graphs via l1 recovery and. Graph structure similarity using spectral graph theory. The observations above tell us that the answer is not nothing.

Spectral graph theory tools for the analysis of complex. The maximum distance between elements of each cluster also called. Two common ways to define distance between two clusters. Spectral graph theory to appear in handbook of linear algebra, second edition, ccr press steve butler fan chungy there are many di erent ways to associate a matrix with a graph an introduction of which. I heard that cvetkovicdoobsachs is supposed to be encyclopedic but i havent had a chance to check it out yet. Spectral graph theory tools for the analysis of complex networks. Virginia commonwealth university vcu scholars compass. Chung, university of pennsylvania, philadelphia, pa.

The improvement ishuge, thanks to theinvaluable comments fromsteve butler, richard stong and many others. Vectors are usually taken to be column vectors unless otherwise speci ed. Lectures on spectral graph theory ucsd mathematics. Spectral graph theory lecture 2 the laplacian daniel a. Algebraic graph theory, by chris godsil and gordon royle. Rittenhouse bachelor of science, university of pittsburgh johnstown, pa 1989 director. Fourier analysis on graphs norbert wiener center for. Lecture notes on expansion, sparsest cut, and spectral graph. Spectral graph theory, by fan chung ucsd mathematics.

I personally think biggs is somewhat dated but more accesible. We have already seen the vertexedge incidence matrix, the laplacian and the adjacency matrix of a graph. For two vertices u and v, the distance between u and v is the number of edges in a shortest path joining u and v. Combining all of the above into a single theorem found in 28, we have. Spectral graph theory to appear in handbook of linear algebra, second edition, ccr press steve butler fan chungy there are many di erent ways to associate a matrix with a graph an introduction of which can be found in chapter 28 on matrices and graphs. Wavelets on graphs via spectral graph theory, applied and computational harmonic analysis 30 2011 no. This is also true in graph theory, and this aspect of graph theory is known as spectral graph theory. The central question of spectral graph theory asks what the spectrum i.

In 1971, fan chung was a promising graduate student at. Source localization on graphs via l1 recovery and spectral graph theory article pdf available march 2016 with 207 reads how we measure reads. It is intuitively related to attempts to understand graphs through the simulation of processes on graphs and through the consideration of physical systems related to graphs. We have already seen the vertexedge incidence matrix, the. Spectral graph theory inthisnoteiusesometerminologiesaboutgraphswithoutde. A good book that is available as a free pdf is spectra of graphs by brouwer and haemers i personally am partial to the normalized laplacian in which case you would want to look at spectral graph theory. Chungs wellwritten exposition can be likened to a conversation with a good teacherone 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. Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. The aim of this book is to understand the spectral grpah theory. Properties and recent applications in spectral graph theory by michelle l.

In this paper, we focus on the connection between the eigenvalues of the laplacian matrix and graph connectivity. Understanding complex networks using graph spectrum. More in particular, spectral graph the ory studies the relation between graph properties and the spectrum of the adjacency matrix or laplace matrix. Spectral graph theory, issue 92 volume 92 of cbms regional conference series issue 92 of regional conference series in mathematics spectral graph theory volume 92 of zebra books. And the theory of association schemes and coherent con. The focus of spectral graph theory is to examine the eigenvalues. The concept of the line graph of a given graph is so natural that it has been independently discovered by many authors. These lecture notes will talk about various matrices which can be associated with a graph, like adjacency. We will start spectral graph theory from these lecture notes.

Another good reference is biggs algebraic graph theory as well as godsil and royles algebraic graph theory same titles, different books. Spectral methods understanding a graph using eigen values and eigen vectors of the matrix we saw. What properties of a graph are exposedrevealed if we 1 represent the graph as. Introduction to spectral graph theory, cheegers inequality are the eigenvalues of ld.