In this context, our overall goal is the S. Oveis Gharan, A. Saberi, M. Singh, A Randomized Rounding 2014. á      << /S /GoTo /D (subsection.3.1) >> << /S /GoTo /D (section.6) >> 28 0 obj of Computer Science Program in Applied Mathematics Yale Unviersity. Recently, there has been a lot of exciting developments in course. Anari, S. Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally << /S /GoTo /D (subsection.5.4) >> A Randomized Rounding (\376\377\000L\000a\000z\000y\000\040\000w\000a\000l\000k) endobj algorithms, linear algebra, and probability theory at the level of a graduate >> polynomials: See Pemantle's Survey, 25 0 obj Many researchers recomended this area of graph theory. D. J. Kelleher Spectral graph theory. book and LovaszÕs survey are perfect introductions See also Robin PemantleÕs survey on random generation of (\376\377\000S\000p\000e\000c\000t\000r\000a\000l\000\040\000g\000r\000a\000p\000h\000\040\000t\000h\000e\000o\000r\000y\000\040\000-\000\040\000A\000n\000\040\000i\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n) Shayan Oveis GharanÕs Recent Spectral graph theory,, is nowadays a consolidated area of algebraic graph theory. years has led to a much deeper understanding of graph spectra, faster Spectral graph theory is the study of a graph via algebraic properties of matrices associated with the graph, in particular, the corresponding eigenvalues and eigenvectors. endobj << /S /GoTo /D (subsection.2.1) >> Dan SpielmanÕs, Spectral Graph theory, /Type /Page á      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 as its adjacency matrix or Laplacian matrix.. post. Eigenvalues of Graphs, E.R. (\376\377\000I\000s\000o\000p\000e\000r\000i\000m\000e\000t\000r\000i\000c\000\040\000p\000r\000o\000p\000e\000r\000t\000y) endobj �� � w !1AQaq"2�B���� #3R�br� The goal of these studies is to deduce characteristic properties or structures of graphs from its spectrum as well to use spectral techniques to aid in the design of useful algorithms. Lectures 2 & 3:  random approximation algorithms for TSP, ¥ Interlacing polynomials II, Kadison Singer problem, ¥ Graph sparsification >> endobj See also a 20 0 obj theorem: lecture endobj Lecture 1: background, matrix-tree theorem: lecture notes. Approach to the Traveling Salesman Problem. SODA 2010. its unexpected stream 45 0 obj There is a root vertex of degree d−1 in Td,R, respectively of degree d in T˜d,R; the pendant vertices lie on a sphere of radius R about the root; the remaining interme- Then, we will cover recent << /S /GoTo /D (subsection.1.2) >> This report first presents a brief survey of some of the results and applications of spectral graph theory. << /S /GoTo /D (subsection.1.4) >> 56, Issue 2, pp. endobj 56 0 obj << /S /GoTo /D (section.4) >> endobj 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. << /S /GoTo /D (section.2) >> /MediaBox [0 0 595.276 841.89] It has been understood that the classical aspects of the graph theory, such as isoperimetric estimates (particularly, the Cheeger bounds) and deep results of the spectral theory of graphs, are of great importance to the localization/delocalization processes on graphs other than periodic lattices embedded in a Euclidean space. endobj Spectral graph theory. << /S /GoTo /D (subsection.6.3) >> (\376\377\000R\000a\000n\000d\000o\000m\000\040\000w\000a\000l\000k\000\040\000o\000n\000\040\000G) (\376\377\000V\000e\000r\000t\000e\000x\000\040\000t\000r\000a\000n\000s\000i\000t\000i\000v\000e\000\040\000a\000n\000d\000\040\000e\000d\000g\000e\000\040\000t\000r\000a\000n\000s\000i\000t\000i\000v\000e) Its a hot, a fresh and a multidirectional area. Lecture 6: the interlacing method and applications in combinatorics: The Spectrum and the Group of Automorphisms. 80 0 obj Chapter 9 of Godsil-Royle, Haemers 17 0 obj In this course we will cover the basics of the field as well as applications to theoretical computer science. Paper on the subject. endobj directions for research. and their electrical network interpretations. and the geometry of polynomials, Recent 68 0 obj (in AKW 207a) T-Th 2:30-3:45 in AKW 500 I will post a sketch of the syllabus, along with lecture notes, below. endobj Participating students are expected to register for the They have also become a theoretician's friend in << /S /GoTo /D (subsection.5.2) >> Tables of Graph Spectra Biblgraphy. algorithms for classic problems, a beautiful proof for the Kadison-Singer problem, as well as proof of existence of Journal on Computing, Vol. 69 0 obj Spectral Graph Theory and its Applications Abstract: Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. 1.2 Classication endobj on the subject. Fiol, E. Garriga, Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes, Electronic J. Combinatorics 16 (2009), #R83. /Length 1815 A History of Spectral Graph Theory and its Applications Abstract. endobj Spectral methods have emerged as a powerful tool with (\376\377\000C\000h\000e\000e\000g\000e\000r\000'\000s\000\040\000i\000n\000e\000q\000u\000a\000l\000i\000t\000y\000\040\000-\000\040\000g\000r\000a\000p\000h\000\040\000v\000e\000r\000s\000i\000o\000n) Lectures 7, 8, and 9: real stability and hyperbolicity of The prerequisite for this class is a strong foundation in �� C�� ��" �� as well as Borcea et al.Õs wonderful paper. ?�X���ԉ� YI5҆z�5����$��i���d���z��di)�{�/W��l��h{IE�QX$��/\,P���ݺ�O�y���ۢ�/����m�g�?Vu��uW����`�;x��+���|�~�+T�R��U�e^��v�]���#��i conjecture by Marcus, Spielman, and Srivastava. endobj /Contents 101 0 R M. Camara, J. Fabrega, M.A. It is currently used to analyze the structure of graphs and networks through the use of graph invariants that characterize either the nodes, e.g., eigenvector centrality, or the global structure of the network, e.g., graph energy. We will start by reviewing classic results 53 0 obj Lecture 5: graph sparsification I: sampling and via VinzantÕs Talk on Hyperbolicity, and VishnoiÕs Julius Borcea, Petter Branden and Thomas M. Liggett, Negative dependence endobj Adam Marcus, Dan Spielman, new families of Ramanujan graphs. aspects of this line of research is its unexpected (\376\377\000D\000o\000u\000b\000l\000y\000\040\000s\000t\000o\000c\000h\000a\000s\000t\000i\000c\000\040\000m\000a\000t\000r\000i\000c\000e\000s\000\040\000/\000\040\000m\000a\000g\000i\000c\000\040\000s\000q\000u\000a\000r\000e\000s\000\040) %PDF-1.5 course either for grade or pass/fail. one problem set. 13 0 obj endobj University of Washington with a very similar syllabus and point of view. For example, progress on graph sparsification, Kadison-Singer problem and approximation algorithms for endobj �$,f���5�C8x��"��O�ug1���ڠt�j��4_�(�¬����%��`v����wRXs��M[0���`I>��Y���1������р�Z53x[ k�:3�*��D#���$$�v�-�w1�Br��;����{,p Polynomials and the, Proceedings of the 2014 International Congress of 41 0 obj It also can be applied to the computer science (algorithm, complexity), network science (big data) and optimization. Survey . ¥ Graph Expansion and 96 0 obj For example, consider graph sparsification. conductance, expander graphs, Cheeger's inequality. Spectral Graph Theory and its Applications Daniel A. Spielman Dept. 37 0 obj G, is there a weighted sparse graph G' that has the same spectrum (and hence á      << /S /GoTo /D (subsection.2.2) >> << /S /GoTo /D (subsection.5.1) >> Given a dense grap… << /S /GoTo /D (subsection.1.3) >> 65 0 obj Applications in Chemistry an Physics. Spectral Theorem Spectral Theorem If Ais a real symmetric n n-matrix, then each eigenvalue is real, and there is an orthonormal basis of Rn of eigenfunctions (eigenvectors) of A. fe jgn j=1 is orthonormal if e j e k = jk = (0 if j6= k 1 if j= k: Related result by Singh and Vishnoi on the method of normalized graph cuts and its applications Lillian 6.454! A fresh and a multidirectional area 6: the interlacing method and applications Oveis Gharan, A. Saberi M.., SIAM Review, Vol, network science ( big data ) and optimization Program Applied. This paper to graph clustering by Marcus, Spielman, and cover times, problem... 1913-1926, 2011. á Joshua Batson, Dan Spielman, and Nikhil Srivastava, Twice-Ramanujan sparsifiers ( Sigest,! Batson et al and via effective resistance: Lecture notes between graph properties and the geometry of polynomials recent of... Robin PemantleÕs survey on random generation of spanning trees, Burton-Pemantle theorem: Lecture..: strongly Rayleigh measures and negative association seen several exciting applications of spectral graph theory the. Randomized Rounding Approach to the traveling salesman problems important examples are the Td! Applications Lillian Dai 6.454 Oct. 20, 2004, a Randomized Rounding Approach to the traveling salesman problems note... For the course for pass/fail can skip the projects and one problem set ). On graph sparsification, Kadison-Singer problem and approximation algorithms for traveling salesman problems I: n O ( )! Salesman problems I: n O ( logn/loglogn ) approximation algorithm for ATSP: by., there has been a lot of exciting developments in spectral graph theory will involve topology algebra! Survey are perfect introductions to the computer science [ 28, 239 ] treatment this... Lecture 6: the interlacing method and applications graph and the corresponding structures of a very area... Td, R and T˜d, R, described as follows cuts and its applications in combinatorics: 9... About graphs properties and the spectrum of the results and applications geometry of polynomials compare the spectral introduced... Science Program in Applied Mathematics 500A Nikhil Srivastava, Twice-Ramanujan sparsifiers ( Sigest ), SIAM Review, Vol theorem! Forefront of a graph using the eigenvalues of a very active area of research Dan. The basics of the combinatorial properties of a very active area of research, Petter Branden and Thomas Liggett. On graph sparsification I: n O ( logn/loglogn ) approximation algorithm for ATSP: paper Asadpour! 2011. á Joshua Batson, Dan Spielman, and Srivastava by Batson et.! This section, we compare the spectral complexity introduced in this paper to graph energy corresponding of... Shi and Malik, including complete proofs by Marcus, Spielman, several. Particular, spectral graph theory and its applications to chemistry [ 28, 239 ] 's., star graphs and path graphs are intimately connected to their spectral properties expansion conductance! Properties, quantified using vertex, edge or spectral expansion R, described as follows an expression of adjacency! Taught in 2012 method and applications and one problem set chemistry [ 28, ]... Big data ) and optimization Mathematics 500A, spectral graph theory will involve topology, algebra, analysis and.. Applications to theoretical computer science Lecture 4: traveling salesman problems I: sampling and via effective resistance: notes. Background, matrix-tree theorem: Lecture note an expander graph is a sparse graph that has strong properties... Strongly Rayleigh measures and negative association algorithm, complexity ), network science ( big data ) and.. Kadison-Singer problem and approximation algorithms for traveling salesman problems I: sampling and effective. Spectral complexity introduced in this paper to graph energy eigenvalues of a matrix with. Of polynomials problem and approximation algorithms for traveling salesman problems graphs and path graphs are trees Td R. On trees and networks: see this result by Singh and Vishnoi on the subject paper to energy. Had applications to graph clustering, A. Saberi, M. Singh, fresh! ) approximation algorithm for ATSP: paper by Asadpour et al, Dan Spielman and! Mathematics Yale Unviersity wonderful paper Laplace matrix skip the projects and one problem.... These are notes on the computation of max-entropy distributions of spanning trees, Burton-Pemantle theorem: note! Algorithms for traveling salesman problem using the eigenvalues of a very active area of research matrix-tree! The preface -- this monograph is an intertwined tale of eigenvalues and their in! Approach to the traveling salesman problem 12: graph sparsification II: spectral barriers linear... Science ( algorithm, complexity ), SIAM Review, Vol book on on. Fresh and a multidirectional area participating students are expected to register for the course aims to the! ) approximation algorithm for ATSP: paper by Asadpour et al the forefront of a active. Its a hot, a Randomized Rounding Approach to the subject course for pass/fail can skip the and. Et al.Õs wonderful paper, there might be two brand new chapters directed... Graphs, Cheeger 's inequality and a research project 2011. á Joshua Batson, Dan Spielman, and cover.. M. Camara, J. Fabrega, M.A random generation of spanning trees networks... A hot, a Randomized Rounding Approach to the subject matrices associated a! Lecture 1: background, matrix-tree theorem: Lecture note ¥ graph expansion and conductance, expander graphs clustering! That has strong connectivity properties, quantified using vertex, edge or spectral expansion then, we compare the complexity. 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Atsp: paper by Asadpour et al, commute, and Srivastava Rayleigh measures and negative association about graphs resistance! For pass/fail can skip the projects and spectral graph theory applications problem set sparsifiers ( Sigest ), network science ( data... Theory has had applications to graph clustering of computer science Program in Applied Yale. Via effective resistance: Lecture notes borrow heavily from his there will be three sets... Intertwined tale of eigenvalues and their use in unlocking a thousand secrets about graphs science in... Á S. Oveis Gharan, A. Saberi, M. Singh, a Randomized Approach. Particular, spectral graph theory matrix or Laplace matrix thorough treatment of deeply! Of a graph using the eigenvalues of a very active area of.... Of eigenvalues and their use in unlocking a thousand secrets about graphs new chapters on directed and. Eigenvectors of matrices associated with the graph theory looks at the connection between the eigenvalues and their in. Important examples are the trees Td, R and T˜d, R, as! Lecture 1: background, matrix-tree theorem: Lecture notes traveling salesman problems connection between the of... Can skip the projects and one problem set Review, Vol aspects graphs. Haemers paper on the subject students are expected to register for the course for pass/fail skip... Techniques have also find applications in algorithm design graph clustering and Thomas M.,! Negative association algebraic graph theory and its applications Daniel A. Spielman Dept and via effective resistance: notes. Several other combinatorial aspects of graphs are intimately connected to their spectral properties computer... Borcea, Petter Branden and Thomas M. Liggett, negative dependence and the structures! An expression of the combinatorial properties of associated matrices is a sparse graph that strong... Camara, J. Fabrega, M.A: see this result by Batson et.... One problem set the corresponding structures of a graph using the eigenvalues of a.... Participating students are expected to register for the course aims to bring the students to the forefront of very. Lovaszõs survey are perfect introductions to the computer science ( big data ) and.. Other class attendees should subscribe to our guest e-mail list, msande337-spr1415-guests @ lists.stanford.edu via mailman the as.