SPECTRAL GRAPH THEORY Contents 1. Introduction 1 2. Basic
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Three Conjectures in Extremal Spectral Graph Theory
Three conjectures in extremal spectral graph theory Michael Tait and Josh Tobin June 6, 2016 Abstract We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetkovi´cand Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and Pn−1. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the K}ovari-S´os-Tur´anTheorem applied to ex (n; K2;t), and a partial solution to an old problem of Erd}oson making a triangle-free graph bipartite. 1 Introduction Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. Perhaps the most well-studied problems in this area are Tur´an-type problems, which ask to maximize the number of edges in a graph which does not contain fixed forbidden subgraphs. Over a century old, a quintessential example of this kind of result is Mantel's theorem, which states that Kdn=2e;bn=2c is the unique graph maximizing the number of edges over all triangle-free graphs. -
Manifold Learning
MANIFOLD LEARNING Kavé Salamatian- LISTIC Learning Machine Learning: develop algorithms to automatically extract ”patterns” or ”regularities” from data (generalization) Typical tasks Clustering: Find groups of similar points Dimensionality reduction: Project points in a lower dimensional space while preserving structure Semi-supervised: Given labelled and unlabelled points, build a labelling function Supervised: Given labelled points, build a labelling function All these tasks are not well-defined Clustering Identify the two groups Dimensionality Reduction 4096 Objects in R ? But only 3 parameters: 2 angles and 1 for illumination. They span a 3- dimensional sub-manifold of R4096! Semi-Supervised Learning Assign labels to black points Supervised learning Build a function that predicts the label of all points in the space Formal definitions Clustering: Given build a function Dimensionality reduction: Given build a function Semi-supervised: Given and with m << n, build a function Supervised: Given , build Geometric learning Consider and exploit the geometric structure of the data Clustering: two ”close” points should be in the same cluster Dimensionality reduction: ”closeness” should be preserved Semi-supervised/supervised: two ”close” points should have the same label Examples: k-means clustering, k-nearest neighbors. Manifold ? A manifold is a topological space that is locally Euclidian Around every point, there is a neighborhood that is topologically isomorph to unit ball Regularization Adopt a functional viewpoint -
Contemporary Spectral Graph Theory
CONTEMPORARY SPECTRAL GRAPH THEORY GUEST EDITORS Maurizio Brunetti, University of Naples Federico II, Italy, [email protected] Raffaella Mulas, The Alan Turing Institute, UK, [email protected] Lucas Rusnak, Texas State University, USA, [email protected] Zoran Stanić, University of Belgrade, Serbia, [email protected] DESCRIPTION Spectral graph theory is a striking theory that studies the properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated with the graph, such as the adjacency matrix, the Laplacian matrix, the signless Laplacian matrix or the distance matrix. This theory emerged in the 1950s and 1960s. Over the decades it has considered not only simple undirected graphs but also their generalizations such as multigraphs, directed graphs, weighted graphs or hypergraphs. In the recent years, spectra of signed graphs and gain graphs have attracted a great deal of attention. Besides graph theoretic research, another major source is the research in a wide branch of applications in chemistry, physics, biology, electrical engineering, social sciences, computer sciences, information theory and other disciplines. This thematic special issue is devoted to the publication of original contributions focusing on the spectra of graphs or their generalizations, along with the most recent applications. Contributions to the Special Issue may address (but are not limited) to the following aspects: f relations between spectra of graphs (or their generalizations) and their structural properties in the most general shape, f bounds for particular eigenvalues, f spectra of particular types of graph, f relations with block designs, f particular eigenvalues of graphs, f cospectrality of graphs, f graph products and their eigenvalues, f graphs with a comparatively small number of eigenvalues, f graphs with particular spectral properties, f applications, f characteristic polynomials. -
Learning on Hypergraphs: Spectral Theory and Clustering
Learning on Hypergraphs: Spectral Theory and Clustering Pan Li, Olgica Milenkovic Coordinated Science Laboratory University of Illinois at Urbana-Champaign March 12, 2019 Learning on Graphs Graphs are indispensable mathematical data models capturing pairwise interactions: k-nn network social network publication network Important learning on graphs problems: clustering (community detection), semi-supervised/active learning, representation learning (graph embedding) etc. Beyond Pairwise Relations A graph models pairwise relations. Recent work has shown that high-order relations can be significantly more informative: Examples include: Understanding the organization of networks (Benson, Gleich and Leskovec'16) Determining the topological connectivity between data points (Zhou, Huang, Sch}olkopf'07). Graphs with high-order relations can be modeled as hypergraphs (formally defined later). Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. High-order network motifs: Motif (Benson’16) Microfauna Pelagic fishes Crabs & Benthic fishes Macroinvertebrates Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. (Zhou, Yu, Han'11) Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Review of Graph Clustering: Notation and Terminology Graph Clustering Task: Cluster the vertices that are \densely" connected by edges. Graph Partitioning and Conductance A (weighted) graph G = (V ; E; w): for e 2 E, we is the weight. -
Course Introduction
Spectral Graph Theory Lecture 1 Introduction Daniel A. Spielman September 2, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. I sometimes edit the notes after class to make them way what I wish I had said. There may be small mistakes, so I recommend that you check any mathematically precise statement before using it in your own work. These notes were last revised on September 3, 2015. 1.1 First Things 1. Please call me \Dan". If such informality makes you uncomfortable, you can try \Professor Dan". If that fails, I will also answer to \Prof. Spielman". 2. If you are going to take this course, please sign up for it on Classes V2. This is the only way you will get emails like \Problem 3 was false, so you don't have to solve it". 3. This class meets this coming Friday, September 4, but not on Labor Day, which is Monday, September 7. 1.2 Introduction I have three objectives in this lecture: to tell you what the course is about, to help you decide if this course is right for you, and to tell you how the course will work. As the title suggests, this course is about the eigenvalues and eigenvectors of matrices associated with graphs, and their applications. I will never forget my amazement at learning that combinatorial properties of graphs could be revealed by an examination of the eigenvalues and eigenvectors of their associated matrices. -
Spectral Geometry for Structural Pattern Recognition
Spectral Geometry for Structural Pattern Recognition HEWAYDA EL GHAWALBY Ph.D. Thesis This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy. Department of Computer Science United Kingdom May 2011 Abstract Graphs are used pervasively in computer science as representations of data with a network or relational structure, where the graph structure provides a flexible representation such that there is no fixed dimensionality for objects. However, the analysis of data in this form has proved an elusive problem; for instance, it suffers from the robustness to structural noise. One way to circumvent this problem is to embed the nodes of a graph in a vector space and to study the properties of the point distribution that results from the embedding. This is a problem that arises in a number of areas including manifold learning theory and graph-drawing. In this thesis, our first contribution is to investigate the heat kernel embed- ding as a route to computing geometric characterisations of graphs. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat ker- nel of the graph is found by exponentiating the Laplacian eigensystem over time. The matrix of embedding co-ordinates for the nodes of the graph is obtained by performing a Young-Householder decomposition on the heat kernel. Once the embedding of its nodes is to hand we proceed to characterise a graph in a geometric manner. With the embeddings to hand, we establish a graph character- ization based on differential geometry by computing sets of curvatures associated ii Abstract iii with the graph nodes, edges and triangular faces. -
CS 598: Spectral Graph Theory. Lecture 10
CS 598: Spectral Graph Theory. Lecture 13 Expander Codes Alexandra Kolla Today Graph approximations, part 2. Bipartite expanders as approximations of the bipartite complete graph Quasi-random properties of bipartite expanders, expander mixing lemma Building expander codes Encoding, minimum distance, decoding Expander Graph Refresher We defined expander graphs to be d- regular graphs whose adjacency matrix eigenvalues satisfy |훼푖| ≤ 휖푑 for i>1, and some small 휖. We saw that such graphs are very good approximations of the complete graph. Expanders as Approximations of the Complete Graph Refresher Let G be a d-regular graph whose adjacency eigenvalues satisfy |훼푖| ≤ 휖푑. As its Laplacian eigenvalues satisfy 휆푖 = 푑 − 훼푖, all non-zero eigenvalues are between (1 − ϵ)d and 1 + ϵ 푑. 푇 푇 Let H=(d/n)Kn, 푠표 푥 퐿퐻푥 = 푑푥 푥 We showed that G is an ϵ − approximation of H 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 Bipartite Expander Graphs Define them as d-regular good approximations of (some multiple of) the complete bipartite 푑 graph H= 퐾 : 푛 푛,푛 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 ⇒ 1 − ϵ 퐻 ≼ 퐺 ≼ 1 + ϵ 퐻 G 퐾푛,푛 U V U V The Spectrum of Bipartite Expanders 푑 Eigenvalues of Laplacian of H= 퐾 are 푛 푛,푛 휆1 = 0, 휆2푛 = 2푑, 휆푖 = 푑, 0 < 푖 < 2푛 푇 푇 푇 1 − ϵ 푥 퐿퐻푥 ≼ 푥 퐿퐺푥 ≼ 1 + ϵ 푥 퐿퐻푥 Says we want d-regular graph G that has evals 휆1 = 0, 휆2푛 = 2푑, 휆푖 ∈ ( 1 − 휖 푑, 1 + 휖 푑), 0 < 푖 < 2푛 Construction of Bipartite Expanders Definition (Double Cover). -
Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science 1-1-2013 Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts Jean H. Gallier University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_reports Part of the Computer Engineering Commons, and the Computer Sciences Commons Recommended Citation Jean H. Gallier, "Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts", . January 2013. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-13-09. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/986 For more information, please contact [email protected]. Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts Abstract These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RP^{N-1}. -
Arxiv:1012.3835V1
A NEW GENERALIZED FIELD OF VALUES RICARDO REIS DA SILVA∗ Abstract. Given a right eigenvector x and a left eigenvector y associated with the same eigen- value of a matrix A, there is a Hermitian positive definite matrix H for which y = Hx. The matrix H defines an inner product and consequently also a field of values. The new generalized field of values is always the convex hull of the eigenvalues of A. Moreover, it is equal to the standard field of values when A is normal and is a particular case of the field of values associated with non-standard inner products proposed by Givens. As a consequence, in the same way as with Hermitian matrices, the eigenvalues of non-Hermitian matrices with real spectrum can be characterized in terms of extrema of a corresponding generalized Rayleigh Quotient. Key words. field of values, numerical range, Rayleigh quotient AMS subject classifications. 15A60, 15A18, 15A42 1. Introduction. We propose a new generalized field of values which brings all the properties of the standard field of values to non-Hermitian matrices. For a matrix A ∈ Cn×n the (classical) field of values or numerical range of A is the set of complex numbers F (A) ≡{x∗Ax : x ∈ Cn, x∗x =1}. (1.1) Alternatively, the field of values of a matrix A ∈ Cn×n can be described as the region in the complex plane defined by the range of the Rayleigh Quotient x∗Ax ρ(x, A) ≡ , ∀x 6=0. (1.2) x∗x For Hermitian matrices, field of values and Rayleigh Quotient exhibit very agreeable properties: F (A) is a subset of the real line and the extrema of F (A) coincide with the extrema of the spectrum, σ(A), of A. -
Spectral Graph Theory: Applications of Courant-Fischer∗
Spectral graph theory: Applications of Courant-Fischer∗ Steve Butler September 2006 Abstract In this second talk we will introduce the Rayleigh quotient and the Courant- Fischer Theorem and give some applications for the normalized Laplacian. Our applications will include structural characterizations of the graph, interlacing results for addition or removal of subgraphs, and interlacing for weak coverings. We also will introduce the idea of “weighted graphs”. 1 Introduction In the first talk we introduced the three common matrices that are used in spectral graph theory, namely the adjacency matrix, the combinatorial Laplacian and the normalized Laplacian. In this talk and the next we will focus on the normalized Laplacian. This is a reflection of the bias of the author having worked closely with Fan Chung who popularized the normalized Laplacian. The adjacency matrix and the combinatorial Laplacian are also very interesting to study in their own right and are useful for giving structural information about a graph. However, in many “practical” applications it tends to be the spectrum of the normalized Laplacian which is most useful. In this talk we will introduce the Rayleigh quotient which is closely linked to the Courant-Fischer Theorem and give some results. We give some structural results (such as showing that if a graph is bipartite then the spectrum is symmetric around 1) as well as some interlacing and weak covering results. But first, we will start by listing some common spectra. ∗Second of three talks given at the Center for Combinatorics, Nankai University, Tianjin 1 Some common spectra When studying spectral graph theory it is good to have a few basic examples to start with and be familiar with their spectrums. -
Spectral Graph Theory – from Practice to Theory
Spectral Graph Theory – From Practice to Theory Alexander Farrugia [email protected] Abstract Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory. Keywords: spectral graph theory, eigenvalue, eigenvector, graph. Introduction: Spectral Graph Theory Graph theory is the area of mathematics that deals with networks. -
Lecture 8: Eigenvalues, Eigenvectors and Spectral Theorem Lecturer: Shayan Oveis Gharan April 20Th Scribe: Sudipto Mukherjee
CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 8: Eigenvalues, Eigenvectors and Spectral Theorem Lecturer: Shayan Oveis Gharan April 20th Scribe: Sudipto Mukherjee Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. 8.1 Introduction Let M 2 Rn×n be a symmetric matrix. We say λ is an eigenvalue of M with eigenvector v, if Mv = λv: Theorem 8.1. If M is symmetric, then all its eigenvalues are real. Proof. Suppose Mv = λv: We want to show that λ has imaginary value 0. For a complex number x = a + ib, the conjugate of x, is defined as follows: x∗ = a − ib. So, all we need to show is that λ = λ∗. The conjugate of a vector is the conjugate of all of its coordinate. Taking the conjugate transpose of both sides of the above equality, we have v∗M = λ∗v∗; (8.1) where we used that M T = M. So, on one hand, v∗Mv = v∗(Mv) = v∗(λv) = λ(v∗v): and on the other hand, by (8.1) v∗Mv = λ∗v∗v: So, we must have λ = λ∗. 8.2 Characteristic Polynomial If M does not have 0 as one of its eigenvalues, then det(M) 6= 0. An equivalent statement is that, if M all columns of M are linearly independent, then det(M) 6= 0. If λ is an eigenvalue of M, then Mv = λv, so (M − λI)v = 0. In other words, M − λI has a zero eigenvalue and det(M − λI) = 0.