/Type /Annot Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. >> endobj /Border[0 0 0]/H/N/C[.5 .5 .5] /Subtype /Link stream /Type /Annot 46 0 obj << 48 0 obj << (References) You are currently offline. /Subtype /Link 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. 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�� /Filter /FlateDecode /Rect [257.302 8.966 264.275 18.431] The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. /Subtype/Link/A<> /Subtype /Link 36 0 obj /A << /S /GoTo /D (Navigation3) >> endobj Network science today is a vast multidisciplinary ﬁeld. 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. 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. 56 0 obj << /Subtype /Link In Proceedings of the 32nd ACM Sym- << /S /GoTo /D (Outline0.8) >> 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. /A << /S /GoTo /D (Navigation1) >> 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. /Subtype /Link endobj (Overview) SPECTRAL GRAPH THEORY (revised and improved) Fan Chung The book was published by AMS in 1992 with a second printing in 1997. /Subtype /Link /A << /S /GoTo /D (Navigation1) >> endobj << /S /GoTo /D (Outline0.1) >> We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. /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 ] Spectral graph theory concerns the connection and interplay between the subjects of graph theory and linear algebra. 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. 32 0 obj (Applications) The common trick we would use to prove stu in spectral graph theory is to decompose the vector into neigenvectors directions. 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. /Type /Annot 51 0 obj << /Type /Annot (Linear Algebra Primer) Important early work was done by social scientists: sociologists, /Type /Annot This problem has been shown to be NP-complete. 19 0 obj endobj >> endobj endobj /Resources 62 0 R 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. Fan R. K. Chung, University of Pennsylvania, Philadelphia, PA. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. However, substantial revision is clearly needed as the list of errata got longer. 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. 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. /Border[0 0 0]/H/N/C[.5 .5 .5] In this paper we begin by introducing basic graph theory terminology. endstream Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. /Font << /F18 65 0 R /F16 66 0 R /F17 67 0 R >> Today, we 35 0 obj /Type /Annot /Type /Annot 55 0 obj << As it turns out, the spectral perspective is a powerful tool. /Subtype /Link Applications in Chemistry an Physics. 50 0 obj << 43 0 obj << 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 $Ł$. /Border[0 0 0]/H/N/C[1 0 0] /Rect [317.389 8.966 328.348 18.431] /Subtype /Link As it turns out, the spectral perspective is a powerful tool. x��VIO1��W�cr��r�R[�*QBnU0�@�L����3�'%��x�����M�(|е���p�F��МX��N��T0�l(��H���Gq��C�mZ�B�cm����=
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q�b!ᬂ{�*�p���U�e ��F�(Ĩ�Ğ���kY ݏ�mp+��$��瓔�95Z�O��� endobj Some features of the site may not work correctly. 39 0 obj The ongoing research in this ﬁeld unravels more and more of them. >> endobj endobj play a major role. >> endobj endobj 8 0 obj << /S /GoTo /D (Outline0.3) >> In the early days, matrix theory and linear algebra … << /S /GoTo /D (Outline0.5) >> (Open Problems) 45 0 obj << endobj Since Gis disconnected, we can split it into two sets Sand Ssuch that jE(S;S)j= 0. /Trans << /S /R >> /Type /Annot /Type /Annot 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. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- /Rect [326.355 8.966 339.307 18.431] Graph analysis provides quantitative tools for the study of complex networks. If M2Cm n /A << /S /GoTo /D (Navigation2) >> Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. 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. The four that in uenced me the most are \Algebraic Graph Theory" by Norman Biggs, v /A << /S /GoTo /D (Navigation1) >> Spectral Graph Theory I Appeared as a branch of algebraic graph theory in the 1950s and 1960s. /Border[0 0 0]/H/N/C[.5 .5 .5] stream It has been found that partitioning a graph based on its spectrum and eigenvectors provides a good /Rect [262.283 8.966 269.257 18.431] 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 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) 64 0 obj << >> endobj The focus of spectral graph theory is … /Border[0 0 0]/H/N/C[.5 .5 .5] 23 0 obj /Contents 63 0 R /A << /S /GoTo /D (Navigation3) >> Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. 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 /Type /Annot /Type /Annot /Subtype /Link /ProcSet [ /PDF /Text ] /Subtype /Link /A << /S /GoTo /D (Navigation2) >> 40 0 obj D. J. Kelleher Spectral graph theory. /Type /Annot Introduction Spectral graph theory has a long history. >> endobj /Filter /FlateDecode Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3. 52 0 obj << Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. 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. 15 0 obj 11 0 obj At ﬁrst glance it might be surprising that such connections exist at all. /Type /Annot /Rect [278.991 8.966 285.965 18.431] /Border[0 0 0]/H/N/C[.5 .5 .5] The Divisor of a Graph. 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.. /A << /S /GoTo /D (Navigation1) >> /Subtype /Link 24 0 obj the theory. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. /A << /S /GoTo /D (Navigation2) >> 61 0 obj <<