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Expander families and Cayley graphs : a beginner's guide /
"The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
New York :
Oxford University Press,
©2011.
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| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 100 | 1 | |a Krebs, Mike. | |
| 245 | 1 | 0 | |a Expander families and Cayley graphs : |b a beginner's guide / |c Mike Krebs and Anthony Shaheen. |
| 260 | |a New York : |b Oxford University Press, |c ©2011. | ||
| 300 | |a 1 online resource (xxiv, 258 pages) : |b illustrations | ||
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| 520 | |a "The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates? Chapters cover topics such as: · Graph spectra · A Cheeger-Buser-type inequality for regular graphs · Group quotients and graph coverings · Subgroups and Schreier generators · Ramanujan graphs and the Alon-Boppana theorem · The zig-zag product and its relation to semidirect products of groups · Representation theory and eigenvalues of Cayley graphs · Kazhdan constants The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects"-- |c Provided by publisher | ||
| 520 | |a "Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects"-- |c Provided by publisher | ||
| 504 | |a Includes bibliographical references (pages 247-252) and index. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Cover; Contents; Preface; Notations and conventions; Introduction; 1. What is an expander family?; 2. What is a Cayley graph?; 3. A tale of four invariants; 4. Applications of expander families; PART ONE: Basics; 1. Graph eigenvalues and the isoperimetric constant; 1. Basic definitions from graph theory; 2. Cayley graphs; 3. The adjacency operator; 4. Eigenvalues of regular graphs; 5. The Laplacian; 6. The isoperimetric constant; 7. The Rayleigh-Ritz theorem; 8. Powers and products of adjacency matrices; 9. An upper bound on the isoperimetric constant; Notes; Exercises. | |
| 590 | |a eBooks on EBSCOhost |b EBSCO eBook Subscription Academic Collection - Worldwide | ||
| 650 | 0 | |a Cayley graphs. | |
| 650 | 0 | |a Eigenvalues. | |
| 650 | 0 | |a Cayley algebras. | |
| 650 | 4 | |a Cayley algebras. | |
| 650 | 4 | |a Cayley graphs. | |
| 650 | 4 | |a Eigenvalues. | |
| 650 | 6 | |a Graphes de Cayley. | |
| 650 | 6 | |a Valeurs propres. | |
| 650 | 6 | |a Algèbres de Cayley. | |
| 650 | 7 | |a MATHEMATICS |x Graphic Methods. |2 bisacsh | |
| 650 | 7 | |a Cayley algebras. |2 fast |0 (OCoLC)fst00849969 | |
| 650 | 7 | |a Cayley graphs. |2 fast |0 (OCoLC)fst00849970 | |
| 650 | 7 | |a Eigenvalues. |2 fast |0 (OCoLC)fst00904031 | |
| 700 | 1 | |a Shaheen, Anthony. | |
| 776 | 0 | 8 | |i Print version: |a Krebs, Mike. |t Expander families and Cayley graphs. |d New York : Oxford University Press, ©2011 |z 9780199767113 |w (DLC) 2011027928 |w (OCoLC)707266118 |
| 856 | 4 | 0 | |u https://ebsco.uam.elogim.com/login.aspx?direct=true&scope=site&db=nlebk&AN=422166 |z Texto completo |
| 880 | 8 | |6 505-00/(S |a 2. Decomposing representations into irreducible representations -- 3. Schur's lemma and characters of representations -- 4. Decomposition of the right regular representation -- 5. Uniqueness of invariant inner products -- 6. Induced representations -- Note -- Exercises -- 7. Representation theory and eigenvalues of Cayley graphs -- 1. Decomposing the adjacency operator into irreps -- 2. Unions of conjugacy classes -- 3. An upper bound on λ(X) -- 4. Eigenvalues of Cayley graphs on abelian groups -- 5. Eigenvalues of Cayley graphs on dihedral groups -- 6. Paley graphs -- Notes -- Exercises -- 8. Kazhdan constants -- 1. Kazhdan constant basics -- 2. The Kazhdan constant, the isoperimetric constant, and the spectral gap -- 3. Abelian groups never yield expander families: A representation-theoretic proof -- 4. Kazhdan constants, subgroups, and quotients -- Notes -- Exercises -- Student research project ideas -- Appendix A: Linear algebra -- 1. Dimension of a vector space -- 2. Inner product spaces, direct sum of subspaces -- 3. The matrix of a linear transformation -- 4. Eigenvalues of linear transformations -- 5. Eigenvalues of circulant matrices -- Appendix B: Asymptotic analysis of functions -- 1. Big oh -- 2. Limit inferior of a function -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Y -- Z. | |
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