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Bogdan Nica Textbooks in Mathematics Bogdan Nica Bogdan Nica A Brief Introduction to A Brief Introduction to Spectral Graph Theory Spectral Graph Theory A Brief Introduction Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to to Spectral Graph structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. Theory This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained. ISBN 978-3-03719-188-0 www.ems-ph.org Nica Cover | Font: Frutiger_Helvetica Neue | Farben: Pantone 116, Pantone 287 | RB 17 mm EMS Textbooks in Mathematics EMS Textbooks in Mathematics is a series of books aimed at students or professional mathemati- cians seeking an introduction into a particular field. The individual volumes are intended not only to provide relevant techniques, results, and applications, but also to afford insight into the motivations and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature. Markus Stroppel, Locally Compact Groups Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations Thomas Timmermann, An Invitation to Quantum Groups and Duality Oleg Bogopolski, Introduction to Group Theory Marek Jarnicki and Peter Pflug, First Steps in Several Complex Variables: Reinhardt Domains Tammo tom Dieck, Algebraic Topology Mauro C. Beltrametti et al., Lectures on Curves, Surfaces and Projective Varieties Wolfgang Woess, Denumerable Markov Chains Eduard Zehnder, Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic Capacities Andrzej Skowro ´nski and Kunio Yamagata, Frobenius Algebras I. Basic Representation Theory Piotr W. Nowak and Guoliang Yu, Large Scale Geometry Joaquim Bruna and Juliá Cufí, Complex Analysis Eduardo Casas-Alvero, Analytic Projective Geometry Fabrice Baudoin, Diffusion Processes and Stochastic Calculus Olivier Lablée, Spectral Theory in Riemannian Geometry Dietmar A. Salamon, Measure and Integration Andrzej Skowro ´nski and Kunio Yamagata, Frobenius Algebras II. Tilted and Hochschild Extension Algebras Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes, Second edition Timothée Marquis, An Introduction to Kac–Moody Groups over Fields Bogdan Nica A Brief Introduction to Spectral Graph Theory Author: Bogdan Nica Department of Mathematics and Statistics McGill University 805 Sherbrooke St W Montreal, QC H3A 0B9 Canada E-mail: [email protected] 2010 Mathematics Subject Classification: Primary: 05-01, 05C50; secondary: 05C25, 11T24, 15A42 Key words: Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs, algebraic graphs over finite fields, character sums ISBN 978-3-03719-188-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2018 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich Switzerland Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 For A. Contents Introduction.................................... 1 1 Graphs.................................... 3 1.1 Notions................................3 1.2 Bipartite graphs............................7 2 Invariants................................... 11 2.1 Chromatic number and independence number............ 11 2.2 Diameter and girth.......................... 12 2.3 Isoperimetric number......................... 15 3 Regular graphs................................ 19 3.1 Cayley graphs............................. 19 3.2 Cayley graphs, continued....................... 22 3.3 Strongly regular graphs........................ 26 3.4 Design graphs............................. 29 4 Finite fields.................................. 31 4.1 Notions................................ 31 4.2 Projective combinatorics....................... 33 4.3 Incidence graphs........................... 36 5 Squares in finite fields............................ 39 5.1 The quadratic character........................ 39 5.2 Quadratic reciprocity......................... 42 5.3 Paley graphs.............................. 44 5.4 A comparison............................. 47 6 Characters.................................. 51 6.1 Characters of finite abelian groups.................. 51 6.2 Character sums over finite fields................... 53 6.3 More character sums over finite fields................ 57 6.4 An application to Paley graphs.................... 59 viii Contents 7 Eigenvalues of graphs............................ 63 7.1 Adjacency and Laplacian eigenvalues................ 63 7.2 First properties............................ 65 7.3 First examples............................. 68 8 Eigenvalue computations........................... 73 8.1 Cayley graphs and bi-Cayley graphs of abelian groups........ 73 8.2 Strongly regular graphs........................ 76 8.3 Two gems............................... 78 8.4 Design graphs............................. 80 9 Largest eigenvalues.............................. 85 9.1 Extremal eigenvalues of symmetric matrices............. 85 9.2 Largest adjacency eigenvalue..................... 86 9.3 The average degree.......................... 88 9.4 A spectral Turán theorem....................... 91 9.5 Largest Laplacian eigenvalue of bipartite graphs........... 93 9.6 Subgraphs............................... 94 9.7 Largest eigenvalues of trees...................... 97 10 More eigenvalues...............................103 10.1 Eigenvalues of symmetric matrices: Courant–Fischer........ 103 10.2 A bound for the Laplacian eigenvalues................ 104 10.3 Eigenvalues of symmetric matrices: Cauchy and Weyl........ 107 10.4 Subgraphs............................... 109 11 Spectral bounds................................113 11.1 Chromatic number and independence number............ 113 11.2 Isoperimetric constant......................... 116 11.3 Edge counting............................. 122 12 Farewell....................................127 12.1 Graphs without 4-cycles....................... 127 12.2 The Erdős–Rényi graph........................ 128 12.3 Eigenvalues of the Erdős–Rényi graph................ 130 Further reading..................................135 Solutions to exercises...............................137 Index.......................................157 Introduction Spectral graph theory starts by associating matrices to graphs — notably, the adja- cency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenval- ues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. To give just one example, spectral ideas are a key ingredient in the proof of the so-called friendship theorem: if, in a group of people, any two persons have exactly one common friend, then there is a person who is everybody’s friend. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. On the one hand, there is, of course, the linear algebra that underlies the spectral ideas in graph theory. On the other hand, most of our examples are graphs of algebraic origin. The two recurring sources are Cayley graphs of groups, and graphs built out of finite fields. In the study of such graphs, some further algebraic ingredients — for example, characters — naturally come up. The table of contents gives, as it should, a good glimpse of where this text is going. Very broadly, the first half is devoted to graphs, finite fields, and how they come together. This part is meant as an appealing and meaningful motivation. It provides a context that frames and fuels much of the second, spectral, half. Many sections have one or two exercises.

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