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...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York :
Oxford University Press,
©2011.
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| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | "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"-- |
| Beschreibung: | 1 online resource (xxiv, 258 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 247-252) and index. |
| ISBN: | 9780199877485 0199877483 9781283427807 128342780X |
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| 245 | 1 | 0 | |a Expander families and Cayley graphs : |b a beginner's guide / |c Mike Krebs and Anthony Shaheen. |
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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. | |
| 650 | 0 | |a Cayley graphs. |0 http://id.loc.gov/authorities/subjects/sh97004851 | |
| 650 | 0 | |a Eigenvalues. |0 http://id.loc.gov/authorities/subjects/sh85041389 | |
| 650 | 0 | |a Cayley algebras. |0 http://id.loc.gov/authorities/subjects/sh85021557 | |
| 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 | |
| 650 | 7 | |a Cayley graphs |2 fast | |
| 650 | 7 | |a Eigenvalues |2 fast | |
| 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 |
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| 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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Datensatz im Suchindex
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn773937205 |
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| _version_ | 1816881783779622912 |
| adam_text | |
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| author | Krebs, Mike |
| author2 | Shaheen, Anthony |
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| author_facet | Krebs, Mike Shaheen, Anthony |
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| callnumber-first | Q - Science |
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| callnumber-raw | QA166.145 .K74 2011eb |
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| collection | ZDB-4-EBA |
| contents | 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. |
| ctrlnum | (OCoLC)773937205 |
| 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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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.</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Coutts Information Services</subfield><subfield code="b">COUT</subfield><subfield code="n">20689143</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">ProQuest Ebook Central</subfield><subfield code="b">EBLB</subfield><subfield code="n">EBL845893</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">ebrary</subfield><subfield code="b">EBRY</subfield><subfield code="n">ebr10524954</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">EBSCOhost</subfield><subfield code="b">EBSC</subfield><subfield code="n">422166</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">ProQuest MyiLibrary Digital eBook Collection</subfield><subfield code="b">IDEB</subfield><subfield code="n">342780</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">YBP Library Services</subfield><subfield code="b">YANK</subfield><subfield code="n">7364148</subfield></datafield><datafield tag="994" ind1=" " ind2=" "><subfield code="a">92</subfield><subfield code="b">GEBAY</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-4-EBA</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-863</subfield></datafield></record></collection> |
| id | ZDB-4-EBA-ocn773937205 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:18:13Z |
| institution | BVB |
| isbn | 9780199877485 0199877483 9781283427807 128342780X |
| language | English |
| oclc_num | 773937205 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xxiv, 258 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Oxford University Press, |
| record_format | marc |
| spelling | Krebs, Mike. Expander families and Cayley graphs : a beginner's guide / Mike Krebs and Anthony Shaheen. New York : Oxford University Press, ©2011. 1 online resource (xxiv, 258 pages) : illustrations text txt rdacontent computer c rdamedia online resource 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"-- Provided by publisher "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"-- Provided by publisher Includes bibliographical references (pages 247-252) and index. Print version record. 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. Cayley graphs. http://id.loc.gov/authorities/subjects/sh97004851 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Cayley algebras. http://id.loc.gov/authorities/subjects/sh85021557 Cayley algebras. Cayley graphs. Eigenvalues. Graphes de Cayley. Valeurs propres. Algèbres de Cayley. MATHEMATICS Graphic Methods. bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast Shaheen, Anthony. Print version: Krebs, Mike. Expander families and Cayley graphs. New York : Oxford University Press, ©2011 9780199767113 (DLC) 2011027928 (OCoLC)707266118 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=422166 Volltext 505-00/(S 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. |
| spellingShingle | Krebs, Mike Expander families and Cayley graphs : a beginner's guide / 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. Cayley graphs. http://id.loc.gov/authorities/subjects/sh97004851 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Cayley algebras. http://id.loc.gov/authorities/subjects/sh85021557 Cayley algebras. Cayley graphs. Eigenvalues. Graphes de Cayley. Valeurs propres. Algèbres de Cayley. MATHEMATICS Graphic Methods. bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh97004851 http://id.loc.gov/authorities/subjects/sh85041389 http://id.loc.gov/authorities/subjects/sh85021557 |
| 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 | Cayley graphs. http://id.loc.gov/authorities/subjects/sh97004851 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Cayley algebras. http://id.loc.gov/authorities/subjects/sh85021557 Cayley algebras. Cayley graphs. Eigenvalues. Graphes de Cayley. Valeurs propres. Algèbres de Cayley. MATHEMATICS Graphic Methods. bisacsh Cayley algebras fast Cayley graphs fast Eigenvalues fast |
| topic_facet | Cayley graphs. Eigenvalues. Cayley algebras. Graphes de Cayley. Valeurs propres. Algèbres de Cayley. MATHEMATICS Graphic Methods. Cayley algebras Cayley graphs Eigenvalues |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=422166 |
| work_keys_str_mv | AT krebsmike expanderfamiliesandcayleygraphsabeginnersguide AT shaheenanthony expanderfamiliesandcayleygraphsabeginnersguide |