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Full Description
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 due to the discovery of Laplace operators that can probe their structure, and their manifold applications from chemical reaction networks to social interactions, they now constitute one of the hottest topics 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 which 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.
Contents
Preface; Acknowledgments; Organization of the book; Part I. Basics: Foundational Material, Elementary Aspects, Examples: 1. Introduction; 2. The abstract setting; 3. The classical case; 4. First properties of the spectrum; 5. Floer theory; Part II. Eigenvalues and Eigenfunctions on Simplicial Complexes and Hypergraphs: 6. Lovász extensions; 7. Discrete p-Laplacians; 8. Cheeger inequalities; 9. Nodal domains; Part III. Additional topics: Interlacing, Tensors, Non-backtracking Laplacians, and Applications: 10. Interlacing and spectral classes; 11. Spectral theory of hypergraphs via tensors; 12. The non-backtracking Laplacian; 13. Applications; Bibliography; Index.
