Expander families and Cayley graphs: a beginner's guide
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York
Oxford University Press
©2011
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| Schlagworte: | |
| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | "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"-- "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"-- Includes bibliographical references (pages 247-252) and index 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 |
| Beschreibung: | 1 Online-Ressource (xxiv, 258 pages) |
| ISBN: | 0199767114 0199877483 9780199767113 9780199877485 |
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| 500 | |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"-- | ||
| 500 | |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"-- | ||
| 500 | |a Includes bibliographical references (pages 247-252) and index | ||
| 500 | |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 | ||
| 650 | 7 | |a MATHEMATICS / Graphic Methods |2 bisacsh | |
| 650 | 7 | |a Cayley algebras |2 fast | |
| 650 | 7 | |a Cayley graphs |2 fast | |
| 650 | 7 | |a Eigenvalues |2 fast | |
| 650 | 4 | |a Cayley graphs | |
| 650 | 4 | |a Eigenvalues | |
| 650 | 4 | |a Cayley algebras | |
| 700 | 1 | |a Shaheen, Anthony |e Sonstige |4 oth | |
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Datensatz im Suchindex
| _version_ | 1804175573505802240 |
|---|---|
| any_adam_object | |
| author | Krebs, Mike |
| author_facet | Krebs, Mike |
| author_role | aut |
| author_sort | Krebs, Mike |
| author_variant | m k mk |
| building | Verbundindex |
| bvnumber | BV043132667 |
| collection | ZDB-4-EBA |
| ctrlnum | (OCoLC)773937205 (DE-599)BVBBV043132667 |
| dewey-full | 511/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.5 |
| dewey-search | 511/.5 |
| dewey-sort | 3511 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | DE-604.BV043132667 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:18:26Z |
| institution | BVB |
| isbn | 0199767114 0199877483 9780199767113 9780199877485 |
| language | English |
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| spelling | Krebs, Mike Verfasser aut Expander families and Cayley graphs a beginner's guide Mike Krebs and Anthony Shaheen New York Oxford University Press ©2011 1 Online-Ressource (xxiv, 258 pages) txt rdacontent c rdamedia cr rdacarrier "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"-- "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"-- Includes bibliographical references (pages 247-252) and index 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 MATHEMATICS / Graphic Methods bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast Cayley graphs Eigenvalues Cayley algebras Shaheen, Anthony Sonstige oth http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=422166 Aggregator Volltext |
| spellingShingle | Krebs, Mike Expander families and Cayley graphs a beginner's guide MATHEMATICS / Graphic Methods bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast Cayley graphs Eigenvalues Cayley algebras |
| title | Expander families and Cayley graphs a beginner's guide |
| title_auth | Expander families and Cayley graphs a beginner's guide |
| title_exact_search | Expander families and Cayley graphs a beginner's guide |
| title_full | Expander families and Cayley graphs a beginner's guide Mike Krebs and Anthony Shaheen |
| title_fullStr | Expander families and Cayley graphs a beginner's guide Mike Krebs and Anthony Shaheen |
| title_full_unstemmed | Expander families and Cayley graphs a beginner's guide Mike Krebs and Anthony Shaheen |
| title_short | Expander families and Cayley graphs |
| title_sort | expander families and cayley graphs a beginner s guide |
| title_sub | a beginner's guide |
| topic | MATHEMATICS / Graphic Methods bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast Cayley graphs Eigenvalues Cayley algebras |
| topic_facet | MATHEMATICS / Graphic Methods Cayley algebras Cayley graphs Eigenvalues |
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