Spectra of Discrete Structures

Jürgen Jost; Raffaella Mulas; Dong Zhang

doi:10.1017/9781009641821

Addressing the active and challenging field of spectral theory, this book develops the general theory of spectra of discrete structures, on graphs, simplicial complexes, and hypergraphs. In fact, hypergraphs have long been neglected in mathematical research, but because of the discovery of Laplace operators that can probe their structure, and their manifold applications from chemical reaction networks to social interactions, they have now become one of the most active areas of interdisciplinary research. The authors' analysis of spectra of discrete structures embeds intuitive and easily visualized examples, which are often quite subtle, within a general mathematical framework. They highlight novel research on Cheeger-type inequalities that connect spectral estimates with the geometry, more precisely the cohesion, of the underlying structure. Establishing mathematical foundations and demonstrating applications, this book will be of interest to graduate students and researchers in mathematics working on the spectral theory of operators on discrete structures.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-viii
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Preface
pp ix-xiii
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Acknowledgments
pp xiv-xiv
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Organization of the Book
pp xv-xviii
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Part I - Basics: Foundational Material, Elementary Aspects, and Examples
pp 1-2
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1 - Introduction
pp 3-13
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2 - The Abstract Setting
pp 14-49
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3 - The Classical Case: Eigenvalues on Euclidean Domains and Riemannian Manifold
pp 50-75
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4 - First Properties of the Spectrum
pp 76-171
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5 - Eigenvalues and Eigenfunctions of the Laplace Operator via Floer Theory
pp 172-178
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Part II - Eigenvalues and Eigenfunctions on Simplicial Complexes and Hypergraphs
pp 179-180
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6 - Lovász Extensions
pp 181-204
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7 - Discrete p-Laplacians
pp 205-231
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8 - Cheeger Inequalities
pp 232-268
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9 - Nodal Domains
pp 269-294
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Part III - Additional Topics: Interlacing, Tensors, Nonbacktracking Laplacians, and Applications
pp 295-296
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10 - Interlacing and Spectral Classes
pp 297-326
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11 - Spectral Theory of Weighted Hypergraphs via Tensors
pp 327-350
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12 - The Nonbacktracking Laplacian
pp 351-372
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13 - Applications
pp 373-387
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Remarks on Notation
pp 388-388
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References
pp 389-402
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Index
pp 403-406
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