�ŖX�n����O޵�������ns�C�b��S'�Y�$��-��F^ې���6�?=t�F�a19���I�.X�5��11i���ҧ�R�N�S�PD�f�����3���k2h������=��em[Blj�%F-8ػ-�.�{&�せ�;O��{�=��Y��c����e��u���Z�Y�1Na����b�Q>�R 41 0 obj << If x= a+ ibis a complex number, then we let x = a ibdenote its conjugate. /Contents 63 0 R 60 0 obj << >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] 59 0 obj << /Parent 70 0 R As it turns out, the spectral perspective is a powerful tool. Network science today is a vast multidisciplinary field. /Rect [267.264 8.966 274.238 18.431] If M2Cm n /Type /Annot Graph analysis provides quantitative tools for the study of complex networks. Some features of the site may not work correctly. @��DoI$�$��`�Q�z0�(4�gp>9~��7����ፇ�lC'��B��#�A�r�4p�Ƣ /A << /S /GoTo /D (Navigation2) >> Download PDF Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. >> endobj /Type /Annot 69 0 obj << (Open Problems) Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. /Subtype /Link /Subtype/Link/A<> /Subtype/Link/A<> /Subtype /Link three topics (Chapters 2{4) in spectral graph theory. /Rect [252.32 8.966 259.294 18.431] >> endobj 8 0 obj /Subtype/Link/A<> 23 0 obj CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. /Type /Annot >> endobj I Research was independently begun in quantum chemistry, as eigenvalues of graphical representation of atoms correspond to energy levels of electrons. We show that in the fine scale limit, for sufficiently regular g , … You are currently offline. (Overview) /A << /S /GoTo /D (Navigation2) >> /Type /Annot >> endobj /Length 794 /Border[0 0 0]/H/N/C[.5 .5 .5] 48 0 obj << 12 0 obj Applications in Chemistry an Physics. /Border[0 0 0]/H/N/C[.5 .5 .5] Lecture 13: Spectral Graph Theory 13-3 Proof. /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] << /S /GoTo /D (Outline0.6) >> 50 0 obj << stream Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. /Rect [352.03 8.966 360.996 18.431] 11.1 Spectral Graph Theory In the eld of spectral graph theory we relate combinatorial properties of graphs to their algebraic properties. endobj /Type /Annot endobj 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.. /Rect [346.052 8.966 354.022 18.431] /Subtype /Link The ongoing research in this field unravels more and more of them. << /S /GoTo /D (Outline0.4) >> /Rect [326.355 8.966 339.307 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] >> endobj 45 0 obj << computational graphs, spectral graph theory, I/O lower bounds ACM Reference Format: Saachi Jain and Matei Zaharia. /Type /Annot /Border[0 0 0]/H/N/C[1 0 0] Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random walks in graph, and to the >> endobj 3.1 Basic de nitions We begin with a brief review of linear algebra. endobj 47 0 obj << 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). 58 0 obj << endobj The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. As it turns out, the spectral perspective is a powerful tool. In this paper we begin by introducing basic graph theory terminology. 43 0 obj << &�r>B������r�a� ����*I��u��1G�`�M�Z0��gb�09f��`��n�B��=�4�8I�sN�"K��*�@�X�IZB��*o����HQ����N�uYY�#�(���T�6s�zgQ%�0�H"�#�Uf;���hvA䔧��q3K*�R�a�J ����h�퀐,���B��P��� 52 0 obj << Because the economy is dynamic and constantly changing, economists should take snapshots of economic data at certain points in time and compare it to other fixed-time data sets to understand 27 0 obj /A << /S /GoTo /D (Navigation1) >> /Border[0 0 0]/H/N/C[.5 .5 .5] In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. << /S /GoTo /D (Outline0.7) >> (Linear Algebra Primer) Appendix. /Subtype /Link /Rect [274.01 8.966 280.984 18.431] /D [41 0 R /XYZ 334.488 0 null] /Rect [244.578 8.966 252.549 18.431] u��KO���s�Mj�E��H��R���'E���I��o8*Y���Sh��e�"")�hb#�.����)�}��|}���[�Bh�}?��X�2!�[email protected]�u�>���h��������.���S��Z���{����x�v8�)1�e3�Ιdc��A������'b[2V�%m��S��M{V�����ط��H�QP�w�����gf=�Bj�)�oE%p�����O�>. /Type /Annot >> endobj Important early work was done by social scientists: sociologists, 104 0 obj << The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. endobj Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. Some Additional Results. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Two important examples are the trees Td,R and T˜d,R, described as follows. Spectral graph theory has proven useful in a number of applications. >> endobj /Rect [295.699 8.966 302.673 18.431] /Rect [300.681 8.966 307.654 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. (Theory) Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. /A << /S /GoTo /D (Navigation3) >> The Spectrum and the Group of Automorphisms. To help the reader reconstruct the ow of my courses, I give three orders that I have used for the material: put orders here There are many terri c books on Spectral Graph Theory. Publication: CBMS Regional Conference Series in Mathematics Publication Year: 1997; Volume 92 ISBNs: 978-0-8218-0315-8 (print); 978-1-4704-2452-7 (online) Characterization of Graphs by Means of Spectra. /Font << /F18 65 0 R /F16 66 0 R /F17 67 0 R >> /Type /Annot In the summer of 2006, the daunting task of revision finally but surely got started. /Subtype /Link /Rect [305.662 8.966 312.636 18.431] >> endobj Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- 36 0 obj In this paper we introduce this spectral graph wavelet transform and study several of its properties. /Border[0 0 0]/H/N/C[1 0 0] /A << /S /GoTo /D (Navigation36) >> /Subtype /Link /Filter /FlateDecode Spectral Graph Theory I Appeared as a branch of algebraic graph theory in the 1950s and 1960s. Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. /Border[0 0 0]/H/N/C[.5 .5 .5] G���&a5�1�S�B}�6�lj[�D��I�Λ&��S��83�b�!�#�t""�b���'�� t�ԫ�nf���B�t�H'��p�m��nY�N2�%~�۽*�m��8s!>�Qю��j��6�9ۥ��~7а��F��|��h ��V�4[��bԦa���zvG�Y�'q�����VԾϒ�K����Έ���Ie��L�k�Q��ΐ�� 62 0 obj << << /S /GoTo /D (Outline0.8) >> Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. 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: 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- endobj >> endobj /Subtype /Link /Type /Annot A major effort in modern graph theory focuses on studying the connection between the eigenvalues of the adjacency matrix of a graph, the graph’s spectrum, and its combinatorial properties. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. In the early days, matrix theory and linear algebra … endobj /Rect [257.302 8.966 264.275 18.431] At each time t, the walk is at /Type /Annot (Applications) /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R /Rect [236.608 8.966 246.571 18.431] Since Gis disconnected, we can split it into two sets Sand Ssuch that jE(S;S)j= 0. << /S /GoTo /D (Outline0.2) >> /Type /Annot >> endobj /A << /S /GoTo /D (Navigation3) >> And the theory of association schemes and coherent con- This problem has been shown to be NP-complete. 64 0 obj << >> endobj << /S /GoTo /D [41 0 R /Fit ] >> /Border[0 0 0]/H/N/C[.5 .5 .5] (Homework Problems) The four that in uenced me the most are \Algebraic Graph Theory" by Norman Biggs, v Introduction Spectral graph theory has a long history. /Filter /FlateDecode /D [41 0 R /XYZ 28.346 272.126 null] /Rect [230.631 8.966 238.601 18.431] >> endobj We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. >> endobj ORIE 6334 Spectral Graph Theory September 22, 2016 Lecture 11 Lecturer: David P. Williamson Scribe: Pu Yang In today’s lecture we will focus on discrete time random walks on undirected graphs. 6 A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY A tree is a graph that has no cycles. Speci cally, we will study random walks on an undirected graph G= (V;E), where the time proceeds in unit steps: t= 1;2;:::. play a major role. << /S /GoTo /D (Outline0.1) >> For instance, star graphs and path graphs are trees. Fan R. K. Chung, University of Pennsylvania, Philadelphia, PA. endobj 42 0 obj << Spectral Graph Theory 5 16.3.2 The Laplacian Quadratic Form Matrices and spectral theory also arise in the study of quadratic forms. In this lecture we discuss Spectral Graph Theory, Conductance, Cheeger’s Inequality, and Spectral Cluster-ing. /Subtype /Link Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. 57 0 obj << /Type /Annot /Length 899 /Rect [288.954 8.966 295.928 18.431] Techniques from spectral graph theory, linear and multilinear algebra, probability, approximation theory, etc. The focus of spectral graph theory is … Lectures #11: Spectral Graph Theory, I Tim Roughgarden & Gregory Valiant May 11, 2020 Spectral graph theory is the powerful and beautiful theory that arises from the following question: What properties of a graph are exposed/revealed if we 1) represent the graph as Let x= 1S j Sj 1S j where as usual 1S represents the indicator of S. The quadratic form of Limplies that xT Lx= 0, as all neighboring vertices were assigned the same weight in x. 24 0 obj /Border[0 0 0]/H/N/C[.5 .5 .5] /Type /Annot 44 0 obj << /A << /S /GoTo /D (Navigation2) >> x��VKO1��W�1���㷏��"!� ɭ�m� )R��o�^B�"PI���[����. 55 0 obj << << /S /GoTo /D (Outline0.5) >> >> >> endobj For instance, extreme eigenvalues of the Laplacian or adjacency matrix are used for partitioning, community detection, dimension reduction for large data sets, data visualization, and a number of other tasks in data science/machine learning theory. At first glance it might be surprising that such connections exist at all. /Rect [339.078 8.966 348.045 18.431] 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. /Type /Annot endobj The most natural quadratic form to associate with a graph is the Laplacian , which is given by xTL Gx = # (a,b)∈E w(a,b)(x(a) −x(b))2. 68 0 obj << /Rect [262.283 8.966 269.257 18.431] 63 0 obj << Spectral Lower Bounds on the I/O Complexity of Computation Graphs. 46 0 obj << (History) endobj /Subtype /Link In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. 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. %PDF-1.4 /Subtype /Link >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] x= X i (fT i x)f i The intuition here is that, we rst compute the projection length of xonto f i which is just the inner product xTf i. /A << /S /GoTo /D (Navigation1) >> %���� 56 0 obj << In Proceedings of the 32nd ACM Sym- /A << /S /GoTo /D (Navigation1) >> /A << /S /GoTo /D (Navigation1) >> /Subtype /Link 39 0 obj endobj endobj endobj /Subtype/Link/A<> Lecture 4 { Spectral Graph Theory Instructors: Geelon So, Nakul Verma Scribes: Jonathan Terry So far, we have studied k-means clustering for nding nice, convex clusters which conform to the standard notion of what a cluster looks like: separated ball-like congregations in space. 53 0 obj << >> endobj /A << /S /GoTo /D (Navigation1) >> We begin with a brief review of linear algebra. 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Network science today is a vast multidisciplinary field. /Rect [267.264 8.966 274.238 18.431] If M2Cm n /Type /Annot Graph analysis provides quantitative tools for the study of complex networks. Some features of the site may not work correctly. @��DoI$�$��`�Q�z0�(4�gp>9~��7����ፇ�lC'��B��#�A�r�4p�Ƣ /A << /S /GoTo /D (Navigation2) >> Download PDF Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. >> endobj /Type /Annot 69 0 obj << (Open Problems) Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. /Subtype /Link /Subtype/Link/A<> /Subtype/Link/A<> /Subtype /Link three topics (Chapters 2{4) in spectral graph theory. /Rect [252.32 8.966 259.294 18.431] >> endobj 8 0 obj /Subtype/Link/A<> 23 0 obj CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. /Type /Annot >> endobj I Research was independently begun in quantum chemistry, as eigenvalues of graphical representation of atoms correspond to energy levels of electrons. We show that in the fine scale limit, for sufficiently regular g , … You are currently offline. (Overview) /A << /S /GoTo /D (Navigation2) >> /Type /Annot >> endobj /Length 794 /Border[0 0 0]/H/N/C[.5 .5 .5] 48 0 obj << 12 0 obj Applications in Chemistry an Physics. /Border[0 0 0]/H/N/C[.5 .5 .5] Lecture 13: Spectral Graph Theory 13-3 Proof. /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] << /S /GoTo /D (Outline0.6) >> 50 0 obj << stream Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. /Rect [352.03 8.966 360.996 18.431] 11.1 Spectral Graph Theory In the eld of spectral graph theory we relate combinatorial properties of graphs to their algebraic properties. endobj /Type /Annot endobj 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.. /Rect [346.052 8.966 354.022 18.431] /Subtype /Link The ongoing research in this field unravels more and more of them. << /S /GoTo /D (Outline0.4) >> /Rect [326.355 8.966 339.307 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] >> endobj 45 0 obj << computational graphs, spectral graph theory, I/O lower bounds ACM Reference Format: Saachi Jain and Matei Zaharia. /Type /Annot /Border[0 0 0]/H/N/C[1 0 0] Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random walks in graph, and to the >> endobj 3.1 Basic de nitions We begin with a brief review of linear algebra. endobj 47 0 obj << 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). 58 0 obj << endobj The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. As it turns out, the spectral perspective is a powerful tool. In this paper we begin by introducing basic graph theory terminology. 43 0 obj << &�r>B������r�a� ����*I��u��1G�`�M�Z0��gb�09f��`��n�B��=�4�8I�sN�"K��*�@�X�IZB��*o����HQ����N�uYY�#�(���T�6s�zgQ%�0�H"�#�Uf;���hvA䔧��q3K*�R�a�J ����h�퀐,���B��P��� 52 0 obj << Because the economy is dynamic and constantly changing, economists should take snapshots of economic data at certain points in time and compare it to other fixed-time data sets to understand 27 0 obj /A << /S /GoTo /D (Navigation1) >> /Border[0 0 0]/H/N/C[.5 .5 .5] In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. << /S /GoTo /D (Outline0.7) >> (Linear Algebra Primer) Appendix. /Subtype /Link /Rect [274.01 8.966 280.984 18.431] /D [41 0 R /XYZ 334.488 0 null] /Rect [244.578 8.966 252.549 18.431] u��KO���s�Mj�E��H��R���'E���I��o8*Y���Sh��e�"")�hb#�.����)�}��|}���[�Bh�}?��X�2!�[email protected]�u�>���h��������.���S��Z���{����x�v8�)1�e3�Ιdc��A������'b[2V�%m��S��M{V�����ط��H�QP�w�����gf=�Bj�)�oE%p�����O�>. /Type /Annot >> endobj Important early work was done by social scientists: sociologists, 104 0 obj << The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. endobj Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. Some Additional Results. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Two important examples are the trees Td,R and T˜d,R, described as follows. Spectral graph theory has proven useful in a number of applications. >> endobj /Rect [295.699 8.966 302.673 18.431] /Rect [300.681 8.966 307.654 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. (Theory) Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. /A << /S /GoTo /D (Navigation3) >> The Spectrum and the Group of Automorphisms. To help the reader reconstruct the ow of my courses, I give three orders that I have used for the material: put orders here There are many terri c books on Spectral Graph Theory. Publication: CBMS Regional Conference Series in Mathematics Publication Year: 1997; Volume 92 ISBNs: 978-0-8218-0315-8 (print); 978-1-4704-2452-7 (online) Characterization of Graphs by Means of Spectra. /Font << /F18 65 0 R /F16 66 0 R /F17 67 0 R >> /Type /Annot In the summer of 2006, the daunting task of revision finally but surely got started. /Subtype /Link /Rect [305.662 8.966 312.636 18.431] >> endobj Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- 36 0 obj In this paper we introduce this spectral graph wavelet transform and study several of its properties. /Border[0 0 0]/H/N/C[1 0 0] /A << /S /GoTo /D (Navigation36) >> /Subtype /Link /Filter /FlateDecode Spectral Graph Theory I Appeared as a branch of algebraic graph theory in the 1950s and 1960s. Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. /Border[0 0 0]/H/N/C[.5 .5 .5] G���&a5�1�S�B}�6�lj[�D��I�Λ&��S��83�b�!�#�t""�b���'�� t�ԫ�nf���B�t�H'��p�m��nY�N2�%~�۽*�m��8s!>�Qю��j��6�9ۥ��~7а��F��|��h ��V�4[��bԦa���zvG�Y�'q�����VԾϒ�K����Έ���Ie��L�k�Q��ΐ�� 62 0 obj << << /S /GoTo /D (Outline0.8) >> Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. 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: 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- endobj >> endobj /Subtype /Link /Type /Annot A major effort in modern graph theory focuses on studying the connection between the eigenvalues of the adjacency matrix of a graph, the graph’s spectrum, and its combinatorial properties. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. In the early days, matrix theory and linear algebra … endobj /Rect [257.302 8.966 264.275 18.431] At each time t, the walk is at /Type /Annot (Applications) /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R /Rect [236.608 8.966 246.571 18.431] Since Gis disconnected, we can split it into two sets Sand Ssuch that jE(S;S)j= 0. << /S /GoTo /D (Outline0.2) >> /Type /Annot >> endobj /A << /S /GoTo /D (Navigation3) >> And the theory of association schemes and coherent con- This problem has been shown to be NP-complete. 64 0 obj << >> endobj << /S /GoTo /D [41 0 R /Fit ] >> /Border[0 0 0]/H/N/C[.5 .5 .5] (Homework Problems) The four that in uenced me the most are \Algebraic Graph Theory" by Norman Biggs, v Introduction Spectral graph theory has a long history. /Filter /FlateDecode /D [41 0 R /XYZ 28.346 272.126 null] /Rect [230.631 8.966 238.601 18.431] >> endobj We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. >> endobj ORIE 6334 Spectral Graph Theory September 22, 2016 Lecture 11 Lecturer: David P. Williamson Scribe: Pu Yang In today’s lecture we will focus on discrete time random walks on undirected graphs. 6 A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY A tree is a graph that has no cycles. Speci cally, we will study random walks on an undirected graph G= (V;E), where the time proceeds in unit steps: t= 1;2;:::. play a major role. << /S /GoTo /D (Outline0.1) >> For instance, star graphs and path graphs are trees. Fan R. K. Chung, University of Pennsylvania, Philadelphia, PA. endobj 42 0 obj << Spectral Graph Theory 5 16.3.2 The Laplacian Quadratic Form Matrices and spectral theory also arise in the study of quadratic forms. In this lecture we discuss Spectral Graph Theory, Conductance, Cheeger’s Inequality, and Spectral Cluster-ing. /Subtype /Link Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. 57 0 obj << /Type /Annot /Length 899 /Rect [288.954 8.966 295.928 18.431] Techniques from spectral graph theory, linear and multilinear algebra, probability, approximation theory, etc. The focus of spectral graph theory is … Lectures #11: Spectral Graph Theory, I Tim Roughgarden & Gregory Valiant May 11, 2020 Spectral graph theory is the powerful and beautiful theory that arises from the following question: What properties of a graph are exposed/revealed if we 1) represent the graph as Let x= 1S j Sj 1S j where as usual 1S represents the indicator of S. The quadratic form of Limplies that xT Lx= 0, as all neighboring vertices were assigned the same weight in x. 24 0 obj /Border[0 0 0]/H/N/C[.5 .5 .5] /Type /Annot 44 0 obj << /A << /S /GoTo /D (Navigation2) >> x��VKO1��W�1���㷏��"!� ɭ�m� )R��o�^B�"PI���[����. 55 0 obj << << /S /GoTo /D (Outline0.5) >> >> >> endobj For instance, extreme eigenvalues of the Laplacian or adjacency matrix are used for partitioning, community detection, dimension reduction for large data sets, data visualization, and a number of other tasks in data science/machine learning theory. At first glance it might be surprising that such connections exist at all. /Rect [339.078 8.966 348.045 18.431] 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. /Type /Annot endobj The most natural quadratic form to associate with a graph is the Laplacian , which is given by xTL Gx = # (a,b)∈E w(a,b)(x(a) −x(b))2. 68 0 obj << /Rect [262.283 8.966 269.257 18.431] 63 0 obj << Spectral Lower Bounds on the I/O Complexity of Computation Graphs. 46 0 obj << (History) endobj /Subtype /Link In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. 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. %PDF-1.4 /Subtype /Link >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] x= X i (fT i x)f i The intuition here is that, we rst compute the projection length of xonto f i which is just the inner product xTf i. /A << /S /GoTo /D (Navigation1) >> %���� 56 0 obj << In Proceedings of the 32nd ACM Sym- /A << /S /GoTo /D (Navigation1) >> /A << /S /GoTo /D (Navigation1) >> /Subtype /Link 39 0 obj endobj endobj endobj /Subtype/Link/A<> Lecture 4 { Spectral Graph Theory Instructors: Geelon So, Nakul Verma Scribes: Jonathan Terry So far, we have studied k-means clustering for nding nice, convex clusters which conform to the standard notion of what a cluster looks like: separated ball-like congregations in space. 53 0 obj << >> endobj /A << /S /GoTo /D (Navigation1) >> We begin with a brief review of linear algebra. 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Network science today is a vast multidisciplinary field. /Rect [267.264 8.966 274.238 18.431] If M2Cm n /Type /Annot Graph analysis provides quantitative tools for the study of complex networks. Some features of the site may not work correctly. @��DoI$�$��`�Q�z0�(4�gp>9~��7����ፇ�lC'��B��#�A�r�4p�Ƣ /A << /S /GoTo /D (Navigation2) >> Download PDF Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. >> endobj /Type /Annot 69 0 obj << (Open Problems) Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. /Subtype /Link /Subtype/Link/A<> /Subtype/Link/A<> /Subtype /Link three topics (Chapters 2{4) in spectral graph theory. /Rect [252.32 8.966 259.294 18.431] >> endobj 8 0 obj /Subtype/Link/A<> 23 0 obj CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. /Type /Annot >> endobj I Research was independently begun in quantum chemistry, as eigenvalues of graphical representation of atoms correspond to energy levels of electrons. We show that in the fine scale limit, for sufficiently regular g , … You are currently offline. (Overview) /A << /S /GoTo /D (Navigation2) >> /Type /Annot >> endobj /Length 794 /Border[0 0 0]/H/N/C[.5 .5 .5] 48 0 obj << 12 0 obj Applications in Chemistry an Physics. /Border[0 0 0]/H/N/C[.5 .5 .5] Lecture 13: Spectral Graph Theory 13-3 Proof. /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] << /S /GoTo /D (Outline0.6) >> 50 0 obj << stream Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. /Rect [352.03 8.966 360.996 18.431] 11.1 Spectral Graph Theory In the eld of spectral graph theory we relate combinatorial properties of graphs to their algebraic properties. endobj /Type /Annot endobj 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.. /Rect [346.052 8.966 354.022 18.431] /Subtype /Link The ongoing research in this field unravels more and more of them. << /S /GoTo /D (Outline0.4) >> /Rect [326.355 8.966 339.307 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] >> endobj 45 0 obj << computational graphs, spectral graph theory, I/O lower bounds ACM Reference Format: Saachi Jain and Matei Zaharia. /Type /Annot /Border[0 0 0]/H/N/C[1 0 0] Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random walks in graph, and to the >> endobj 3.1 Basic de nitions We begin with a brief review of linear algebra. endobj 47 0 obj << 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). 58 0 obj << endobj The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. As it turns out, the spectral perspective is a powerful tool. In this paper we begin by introducing basic graph theory terminology. 43 0 obj << &�r>B������r�a� ����*I��u��1G�`�M�Z0��gb�09f��`��n�B��=�4�8I�sN�"K��*�@�X�IZB��*o����HQ����N�uYY�#�(���T�6s�zgQ%�0�H"�#�Uf;���hvA䔧��q3K*�R�a�J ����h�퀐,���B��P��� 52 0 obj << Because the economy is dynamic and constantly changing, economists should take snapshots of economic data at certain points in time and compare it to other fixed-time data sets to understand 27 0 obj /A << /S /GoTo /D (Navigation1) >> /Border[0 0 0]/H/N/C[.5 .5 .5] In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. << /S /GoTo /D (Outline0.7) >> (Linear Algebra Primer) Appendix. /Subtype /Link /Rect [274.01 8.966 280.984 18.431] /D [41 0 R /XYZ 334.488 0 null] /Rect [244.578 8.966 252.549 18.431] u��KO���s�Mj�E��H��R���'E���I��o8*Y���Sh��e�"")�hb#�.����)�}��|}���[�Bh�}?��X�2!�[email protected]�u�>���h��������.���S��Z���{����x�v8�)1�e3�Ιdc��A������'b[2V�%m��S��M{V�����ط��H�QP�w�����gf=�Bj�)�oE%p�����O�>. /Type /Annot >> endobj Important early work was done by social scientists: sociologists, 104 0 obj << The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. endobj Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. Some Additional Results. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Two important examples are the trees Td,R and T˜d,R, described as follows. Spectral graph theory has proven useful in a number of applications. >> endobj /Rect [295.699 8.966 302.673 18.431] /Rect [300.681 8.966 307.654 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. (Theory) Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. /A << /S /GoTo /D (Navigation3) >> The Spectrum and the Group of Automorphisms. To help the reader reconstruct the ow of my courses, I give three orders that I have used for the material: put orders here There are many terri c books on Spectral Graph Theory. Publication: CBMS Regional Conference Series in Mathematics Publication Year: 1997; Volume 92 ISBNs: 978-0-8218-0315-8 (print); 978-1-4704-2452-7 (online) Characterization of Graphs by Means of Spectra. /Font << /F18 65 0 R /F16 66 0 R /F17 67 0 R >> /Type /Annot In the summer of 2006, the daunting task of revision finally but surely got started. /Subtype /Link /Rect [305.662 8.966 312.636 18.431] >> endobj Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- 36 0 obj In this paper we introduce this spectral graph wavelet transform and study several of its properties. /Border[0 0 0]/H/N/C[1 0 0] /A << /S /GoTo /D (Navigation36) >> /Subtype /Link /Filter /FlateDecode Spectral Graph Theory I Appeared as a branch of algebraic graph theory in the 1950s and 1960s. Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. /Border[0 0 0]/H/N/C[.5 .5 .5] G���&a5�1�S�B}�6�lj[�D��I�Λ&��S��83�b�!�#�t""�b���'�� t�ԫ�nf���B�t�H'��p�m��nY�N2�%~�۽*�m��8s!>�Qю��j��6�9ۥ��~7а��F��|��h ��V�4[��bԦa���zvG�Y�'q�����VԾϒ�K����Έ���Ie��L�k�Q��ΐ�� 62 0 obj << << /S /GoTo /D (Outline0.8) >> Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. 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: 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- endobj >> endobj /Subtype /Link /Type /Annot A major effort in modern graph theory focuses on studying the connection between the eigenvalues of the adjacency matrix of a graph, the graph’s spectrum, and its combinatorial properties. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. In the early days, matrix theory and linear algebra … endobj /Rect [257.302 8.966 264.275 18.431] At each time t, the walk is at /Type /Annot (Applications) /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R /Rect [236.608 8.966 246.571 18.431] Since Gis disconnected, we can split it into two sets Sand Ssuch that jE(S;S)j= 0. << /S /GoTo /D (Outline0.2) >> /Type /Annot >> endobj /A << /S /GoTo /D (Navigation3) >> And the theory of association schemes and coherent con- This problem has been shown to be NP-complete. 64 0 obj << >> endobj << /S /GoTo /D [41 0 R /Fit ] >> /Border[0 0 0]/H/N/C[.5 .5 .5] (Homework Problems) The four that in uenced me the most are \Algebraic Graph Theory" by Norman Biggs, v Introduction Spectral graph theory has a long history. /Filter /FlateDecode /D [41 0 R /XYZ 28.346 272.126 null] /Rect [230.631 8.966 238.601 18.431] >> endobj We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. >> endobj ORIE 6334 Spectral Graph Theory September 22, 2016 Lecture 11 Lecturer: David P. Williamson Scribe: Pu Yang In today’s lecture we will focus on discrete time random walks on undirected graphs. 6 A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY A tree is a graph that has no cycles. Speci cally, we will study random walks on an undirected graph G= (V;E), where the time proceeds in unit steps: t= 1;2;:::. play a major role. << /S /GoTo /D (Outline0.1) >> For instance, star graphs and path graphs are trees. Fan R. K. Chung, University of Pennsylvania, Philadelphia, PA. endobj 42 0 obj << Spectral Graph Theory 5 16.3.2 The Laplacian Quadratic Form Matrices and spectral theory also arise in the study of quadratic forms. In this lecture we discuss Spectral Graph Theory, Conductance, Cheeger’s Inequality, and Spectral Cluster-ing. /Subtype /Link Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. 57 0 obj << /Type /Annot /Length 899 /Rect [288.954 8.966 295.928 18.431] Techniques from spectral graph theory, linear and multilinear algebra, probability, approximation theory, etc. The focus of spectral graph theory is … Lectures #11: Spectral Graph Theory, I Tim Roughgarden & Gregory Valiant May 11, 2020 Spectral graph theory is the powerful and beautiful theory that arises from the following question: What properties of a graph are exposed/revealed if we 1) represent the graph as Let x= 1S j Sj 1S j where as usual 1S represents the indicator of S. The quadratic form of Limplies that xT Lx= 0, as all neighboring vertices were assigned the same weight in x. 24 0 obj /Border[0 0 0]/H/N/C[.5 .5 .5] /Type /Annot 44 0 obj << /A << /S /GoTo /D (Navigation2) >> x��VKO1��W�1���㷏��"!� ɭ�m� )R��o�^B�"PI���[����. 55 0 obj << << /S /GoTo /D (Outline0.5) >> >> >> endobj For instance, extreme eigenvalues of the Laplacian or adjacency matrix are used for partitioning, community detection, dimension reduction for large data sets, data visualization, and a number of other tasks in data science/machine learning theory. At first glance it might be surprising that such connections exist at all. /Rect [339.078 8.966 348.045 18.431] 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. /Type /Annot endobj The most natural quadratic form to associate with a graph is the Laplacian , which is given by xTL Gx = # (a,b)∈E w(a,b)(x(a) −x(b))2. 68 0 obj << /Rect [262.283 8.966 269.257 18.431] 63 0 obj << Spectral Lower Bounds on the I/O Complexity of Computation Graphs. 46 0 obj << (History) endobj /Subtype /Link In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. 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. %PDF-1.4 /Subtype /Link >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] x= X i (fT i x)f i The intuition here is that, we rst compute the projection length of xonto f i which is just the inner product xTf i. /A << /S /GoTo /D (Navigation1) >> %���� 56 0 obj << In Proceedings of the 32nd ACM Sym- /A << /S /GoTo /D (Navigation1) >> /A << /S /GoTo /D (Navigation1) >> /Subtype /Link 39 0 obj endobj endobj endobj /Subtype/Link/A<> Lecture 4 { Spectral Graph Theory Instructors: Geelon So, Nakul Verma Scribes: Jonathan Terry So far, we have studied k-means clustering for nding nice, convex clusters which conform to the standard notion of what a cluster looks like: separated ball-like congregations in space. 53 0 obj << >> endobj /A << /S /GoTo /D (Navigation1) >> We begin with a brief review of linear algebra. 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>> 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. To give just one example, spectral…, The adjacency algebra of a graph, with an application to affine planes, Approximate graph spectral decomposition with the Variational Quantum Eigensolver, Some results on the Laplacian Spread Conjecture, Volume of Seifert representations for graph manifolds and their finite covers, On the spectrum of an equitable quotient matrix and its application, Spectral Graph Analysis with Apache Spark, Spectrum of some arrow-bordered circulant matrix, Geometric Formulation for Discrete Points and its Applications, I ’ ve got 99 vertices but a solution to Conway ’ s problem ain ’ t one, Polaritons and excitons: Hamiltonian design for enhanced coherence, By clicking accept or continuing to use the site, you agree to the terms outlined in our. 11 0 obj endstream /Trans << /S /R >> Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. 28 0 obj 32 0 obj 2020. Spectra Techniques in Graph Theory and Combinatories. /Type /Annot /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] >> /Border[0 0 0]/H/N/C[.5 .5 .5] Spectral graph theory Economics is a social science that tries to understand how supply and demand control the allocation of limited resources. (16.2) This form measures the smoothness of the function x. 49 0 obj << 35 0 obj More in particular, spectral graph the-ory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. I Early work focused on using the adjacency matrix, which limited initial results to regular graphs. /A << /S /GoTo /D (Navigation2) >> In Chapter1, we review the basic de nitions, notations, and results in graph theory and spectral graph theory. endobj 54 0 obj << /Subtype /Link x��VIO1��W�cr��r�R[�*QBnU0�@�L����3�'%��x�����M�(|е���p�F��МX��N��T0�l(��H���Gq��C�mZ�B�cm����= >}\0��ƈT�zp � q�b!ᬂ{�*�p���U�e ��F�(Ĩ�Ğ���kY ݏ�mp+��$��瓔�95Z�O��� stream The wide range of these topics showcases the power and versatility of the eigenvalue techniques such as interlacing, the common thread that ties these topics together. SPECTRAL GRAPH THEORY (revised and improved) Fan Chung The book was published by AMS in 1992 with a second printing in 1997. /Border[0 0 0]/H/N/C[1 0 0] /Subtype /Link /Subtype /Link Spectral Graph Theory About this Title. 20 0 obj 19 0 obj >> endobj Our applications will include structural characterizations of the graph, interlacing /D [41 0 R /XYZ 334.488 0 null] >> endobj spectral techniques in solving graph partitioning problems where graph vertices are partitioned into two disjoint sets of similar sizes while the number of edges between the two sets is minimized. /Border[0 0 0]/H/N/C[.5 .5 .5] >> endobj Then we multiply … /Type /Annot 40 0 obj /Type /Annot /A << /S /GoTo /D (Navigation1) >> /Type /Annot the theory. /A << /S /GoTo /D (Navigation1) >> Introduction to Spectral Graph Theory Spectral graph theory is the study of a graph through the properties of the eigenvalues and eigenvectors of its associated Laplacian matrix. /Border[0 0 0]/H/N/C[1 0 0] /A << /S /GoTo /D (Navigation1) >> (References) The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. The Divisor of a Graph. >> endobj 15 0 obj Spectral graph drawing: Tutte justification Gives for all i λsmall says x(i) near average of neighbors Tutte ‘63: If fix outside face, and let every other vertex be average of neighbors, get planar embedding of planar graph. /Resources 62 0 R << /S /GoTo /D (Outline0.3) >> 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. endobj D. J. Kelleher Spectral graph theory. /Rect [310.643 8.966 317.617 18.431] Today, we /Type /Annot >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] endobj Tables of Graph Spectra Biblgraphy. /Rect [317.389 8.966 328.348 18.431] /Rect [283.972 8.966 290.946 18.431] /Annots [ 42 0 R 43 0 R 44 0 R 45 0 R 46 0 R 47 0 R 48 0 R 49 0 R 50 0 R 51 0 R 52 0 R 53 0 R 54 0 R 55 0 R 56 0 R 57 0 R 58 0 R 59 0 R 60 0 R 61 0 R ] /MediaBox [0 0 362.835 272.126] 31 0 obj /ProcSet [ /PDF /Text ] Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. 51 0 obj << The common trick we would use to prove stu in spectral graph theory is to decompose the vector into neigenvectors directions. /Type /Page endobj >> endobj 16 0 obj /Rect [278.991 8.966 285.965 18.431] endobj 61 0 obj << Spectral graph theory concerns the connection and interplay between the subjects of graph theory and linear algebra. Relations Between Spectral and Structural Properties of Graphs. S���r�/STz�|eU���–Jڤ"�W�t� m�H�bt�o�#�H}l��͂^��./����g��Dz?����7^���m���d���-g�|�w����6�����)�U�,]Ut�qLYH���l��DE����ȕB,�\��A��i��L�S��C�}�B���x�J�j��7'������+����J����X�R��"�YA|���ݖ=�f=>�ŖX�n����O޵�������ns�C�b��S'�Y�$��-��F^ې���6�?=t�F�a19���I�.X�5��11i���ҧ�R�N�S�PD�f�����3���k2h������=��em[Blj�%F-8ػ-�.�{&�せ�;O��{�=��Y��c����e��u���Z�Y�1Na����b�Q>�R 41 0 obj << If x= a+ ibis a complex number, then we let x = a ibdenote its conjugate. /Contents 63 0 R 60 0 obj << >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] 59 0 obj << /Parent 70 0 R As it turns out, the spectral perspective is a powerful tool. Network science today is a vast multidisciplinary field. /Rect [267.264 8.966 274.238 18.431] If M2Cm n /Type /Annot Graph analysis provides quantitative tools for the study of complex networks. Some features of the site may not work correctly. @��DoI$�$��`�Q�z0�(4�gp>9~��7����ፇ�lC'��B��#�A�r�4p�Ƣ /A << /S /GoTo /D (Navigation2) >> Download PDF Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. >> endobj /Type /Annot 69 0 obj << (Open Problems) Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. /Subtype /Link /Subtype/Link/A<> /Subtype/Link/A<> /Subtype /Link three topics (Chapters 2{4) in spectral graph theory. /Rect [252.32 8.966 259.294 18.431] >> endobj 8 0 obj /Subtype/Link/A<> 23 0 obj CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. /Type /Annot >> endobj I Research was independently begun in quantum chemistry, as eigenvalues of graphical representation of atoms correspond to energy levels of electrons. We show that in the fine scale limit, for sufficiently regular g , … You are currently offline. (Overview) /A << /S /GoTo /D (Navigation2) >> /Type /Annot >> endobj /Length 794 /Border[0 0 0]/H/N/C[.5 .5 .5] 48 0 obj << 12 0 obj Applications in Chemistry an Physics. /Border[0 0 0]/H/N/C[.5 .5 .5] Lecture 13: Spectral Graph Theory 13-3 Proof. /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] << /S /GoTo /D (Outline0.6) >> 50 0 obj << stream Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. /Rect [352.03 8.966 360.996 18.431] 11.1 Spectral Graph Theory In the eld of spectral graph theory we relate combinatorial properties of graphs to their algebraic properties. endobj /Type /Annot endobj 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.. /Rect [346.052 8.966 354.022 18.431] /Subtype /Link The ongoing research in this field unravels more and more of them. << /S /GoTo /D (Outline0.4) >> /Rect [326.355 8.966 339.307 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] >> endobj 45 0 obj << computational graphs, spectral graph theory, I/O lower bounds ACM Reference Format: Saachi Jain and Matei Zaharia. /Type /Annot /Border[0 0 0]/H/N/C[1 0 0] Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random walks in graph, and to the >> endobj 3.1 Basic de nitions We begin with a brief review of linear algebra. endobj 47 0 obj << 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). 58 0 obj << endobj The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. As it turns out, the spectral perspective is a powerful tool. In this paper we begin by introducing basic graph theory terminology. 43 0 obj << &�r>B������r�a� ����*I��u��1G�`�M�Z0��gb�09f��`��n�B��=�4�8I�sN�"K��*�@�X�IZB��*o����HQ����N�uYY�#�(���T�6s�zgQ%�0�H"�#�Uf;���hvA䔧��q3K*�R�a�J ����h�퀐,���B��P��� 52 0 obj << Because the economy is dynamic and constantly changing, economists should take snapshots of economic data at certain points in time and compare it to other fixed-time data sets to understand 27 0 obj /A << /S /GoTo /D (Navigation1) >> /Border[0 0 0]/H/N/C[.5 .5 .5] In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. << /S /GoTo /D (Outline0.7) >> (Linear Algebra Primer) Appendix. /Subtype /Link /Rect [274.01 8.966 280.984 18.431] /D [41 0 R /XYZ 334.488 0 null] /Rect [244.578 8.966 252.549 18.431] u��KO���s�Mj�E��H��R���'E���I��o8*Y���Sh��e�"")�hb#�.����)�}��|}���[�Bh�}?��X�2!�[email protected]�u�>���h��������.���S��Z���{����x�v8�)1�e3�Ιdc��A������'b[2V�%m��S��M{V�����ط��H�QP�w�����gf=�Bj�)�oE%p�����O�>. /Type /Annot >> endobj Important early work was done by social scientists: sociologists, 104 0 obj << The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. endobj Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. Some Additional Results. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Two important examples are the trees Td,R and T˜d,R, described as follows. Spectral graph theory has proven useful in a number of applications. >> endobj /Rect [295.699 8.966 302.673 18.431] /Rect [300.681 8.966 307.654 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. (Theory) Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. /A << /S /GoTo /D (Navigation3) >> The Spectrum and the Group of Automorphisms. To help the reader reconstruct the ow of my courses, I give three orders that I have used for the material: put orders here There are many terri c books on Spectral Graph Theory. Publication: CBMS Regional Conference Series in Mathematics Publication Year: 1997; Volume 92 ISBNs: 978-0-8218-0315-8 (print); 978-1-4704-2452-7 (online) Characterization of Graphs by Means of Spectra. /Font << /F18 65 0 R /F16 66 0 R /F17 67 0 R >> /Type /Annot In the summer of 2006, the daunting task of revision finally but surely got started. /Subtype /Link /Rect [305.662 8.966 312.636 18.431] >> endobj Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- 36 0 obj In this paper we introduce this spectral graph wavelet transform and study several of its properties. /Border[0 0 0]/H/N/C[1 0 0] /A << /S /GoTo /D (Navigation36) >> /Subtype /Link /Filter /FlateDecode Spectral Graph Theory I Appeared as a branch of algebraic graph theory in the 1950s and 1960s. Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. /Border[0 0 0]/H/N/C[.5 .5 .5] G���&a5�1�S�B}�6�lj[�D��I�Λ&��S��83�b�!�#�t""�b���'�� t�ԫ�nf���B�t�H'��p�m��nY�N2�%~�۽*�m��8s!>�Qю��j��6�9ۥ��~7а��F��|��h ��V�4[��bԦa���zvG�Y�'q�����VԾϒ�K����Έ���Ie��L�k�Q��ΐ�� 62 0 obj << << /S /GoTo /D (Outline0.8) >> Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. 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: 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- endobj >> endobj /Subtype /Link /Type /Annot A major effort in modern graph theory focuses on studying the connection between the eigenvalues of the adjacency matrix of a graph, the graph’s spectrum, and its combinatorial properties. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. In the early days, matrix theory and linear algebra … endobj /Rect [257.302 8.966 264.275 18.431] At each time t, the walk is at /Type /Annot (Applications) /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R /Rect [236.608 8.966 246.571 18.431] Since Gis disconnected, we can split it into two sets Sand Ssuch that jE(S;S)j= 0. << /S /GoTo /D (Outline0.2) >> /Type /Annot >> endobj /A << /S /GoTo /D (Navigation3) >> And the theory of association schemes and coherent con- This problem has been shown to be NP-complete. 64 0 obj << >> endobj << /S /GoTo /D [41 0 R /Fit ] >> /Border[0 0 0]/H/N/C[.5 .5 .5] (Homework Problems) The four that in uenced me the most are \Algebraic Graph Theory" by Norman Biggs, v Introduction Spectral graph theory has a long history. /Filter /FlateDecode /D [41 0 R /XYZ 28.346 272.126 null] /Rect [230.631 8.966 238.601 18.431] >> endobj We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. >> endobj ORIE 6334 Spectral Graph Theory September 22, 2016 Lecture 11 Lecturer: David P. Williamson Scribe: Pu Yang In today’s lecture we will focus on discrete time random walks on undirected graphs. 6 A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY A tree is a graph that has no cycles. Speci cally, we will study random walks on an undirected graph G= (V;E), where the time proceeds in unit steps: t= 1;2;:::. play a major role. << /S /GoTo /D (Outline0.1) >> For instance, star graphs and path graphs are trees. Fan R. K. Chung, University of Pennsylvania, Philadelphia, PA. endobj 42 0 obj << Spectral Graph Theory 5 16.3.2 The Laplacian Quadratic Form Matrices and spectral theory also arise in the study of quadratic forms. In this lecture we discuss Spectral Graph Theory, Conductance, Cheeger’s Inequality, and Spectral Cluster-ing. /Subtype /Link Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. 57 0 obj << /Type /Annot /Length 899 /Rect [288.954 8.966 295.928 18.431] Techniques from spectral graph theory, linear and multilinear algebra, probability, approximation theory, etc. The focus of spectral graph theory is … Lectures #11: Spectral Graph Theory, I Tim Roughgarden & Gregory Valiant May 11, 2020 Spectral graph theory is the powerful and beautiful theory that arises from the following question: What properties of a graph are exposed/revealed if we 1) represent the graph as Let x= 1S j Sj 1S j where as usual 1S represents the indicator of S. The quadratic form of Limplies that xT Lx= 0, as all neighboring vertices were assigned the same weight in x. 24 0 obj /Border[0 0 0]/H/N/C[.5 .5 .5] /Type /Annot 44 0 obj << /A << /S /GoTo /D (Navigation2) >> x��VKO1��W�1���㷏��"!� ɭ�m� )R��o�^B�"PI���[����. 55 0 obj << << /S /GoTo /D (Outline0.5) >> >> >> endobj For instance, extreme eigenvalues of the Laplacian or adjacency matrix are used for partitioning, community detection, dimension reduction for large data sets, data visualization, and a number of other tasks in data science/machine learning theory. At first glance it might be surprising that such connections exist at all. /Rect [339.078 8.966 348.045 18.431] 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. /Type /Annot endobj The most natural quadratic form to associate with a graph is the Laplacian , which is given by xTL Gx = # (a,b)∈E w(a,b)(x(a) −x(b))2. 68 0 obj << /Rect [262.283 8.966 269.257 18.431] 63 0 obj << Spectral Lower Bounds on the I/O Complexity of Computation Graphs. 46 0 obj << (History) endobj /Subtype /Link In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. 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. %PDF-1.4 /Subtype /Link >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] x= X i (fT i x)f i The intuition here is that, we rst compute the projection length of xonto f i which is just the inner product xTf i. /A << /S /GoTo /D (Navigation1) >> %���� 56 0 obj << In Proceedings of the 32nd ACM Sym- /A << /S /GoTo /D (Navigation1) >> /A << /S /GoTo /D (Navigation1) >> /Subtype /Link 39 0 obj endobj endobj endobj /Subtype/Link/A<> Lecture 4 { Spectral Graph Theory Instructors: Geelon So, Nakul Verma Scribes: Jonathan Terry So far, we have studied k-means clustering for nding nice, convex clusters which conform to the standard notion of what a cluster looks like: separated ball-like congregations in space. 53 0 obj << >> endobj /A << /S /GoTo /D (Navigation1) >> We begin with a brief review of linear algebra. 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