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Expander families and Cayley graphs [[electronic resource] ] : a beginner's guide / / Mike Krebs and Anthony Shaheen
| Expander families and Cayley graphs [[electronic resource] ] : a beginner's guide / / Mike Krebs and Anthony Shaheen |
| Autore | Krebs Mike |
| Pubbl/distr/stampa | Oxford ; ; New York, : Oxford University Press, c2011 |
| Descrizione fisica | 1 online resource (283 p.) |
| Disciplina | 511/.5 |
| Altri autori (Persone) | ShaheenAnthony |
| Soggetto topico |
Cayley graphs
Eigenvalues Cayley algebras |
| Soggetto genere / forma | Electronic books. |
| ISBN |
1-283-42780-X
9786613427809 0-19-987748-3 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
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
2. Subgroups and quotients1. Coverings and quotients; 2. Subgroups and Schreier generators; Notes; Exercises; Student research project ideas; 3. The Alon-Boppana theorem; 1. Statement and consequences; 2. First proof: The Rayleigh-Ritz method; 3. Second proof: The trace method; Notes; Exercises; Student research project ideas; PART TWO: Combinatorial Techniques; 4. Diameters of Cayley graphs and expander families; 1. Expander families have logarithmic diameter; 2. Diameters of Cayley graphs; 3. Abelian groups never yield expander families: A combinatorial proof 4. Diameters of subgroups and quotients5. Solvable groups with bounded derived length; 6. Semidirect products and wreath products; 7. Cube-connected cycle graphs; Notes; Exercises; Student research project ideas; 5. Zig-zag products; 1. Definition of the zig-zag product; 2. Adjacency matrices and zig-zag products; 3. Eigenvalues of zig-zag products; 4. An actual expander family; 5. Zig-zag products and semidirect products; Notes; Exercises; Student research project ideas; PART THREE: Representation-Theoretic Techniques; 6. Representations of finite groups; 1. Representations of finite groups 2. Decomposing representations into irreducible representations3. 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 ?(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 basics2. 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 |
| Record Nr. | UNINA-9910457774803321 |
Krebs Mike
|
||
| Oxford ; ; New York, : Oxford University Press, c2011 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Expander families and Cayley graphs [[electronic resource] ] : a beginner's guide / / Mike Krebs and Anthony Shaheen
| Expander families and Cayley graphs [[electronic resource] ] : a beginner's guide / / Mike Krebs and Anthony Shaheen |
| Autore | Krebs Mike |
| Pubbl/distr/stampa | Oxford ; ; New York, : Oxford University Press, c2011 |
| Descrizione fisica | 1 online resource (283 p.) |
| Disciplina | 511/.5 |
| Altri autori (Persone) | ShaheenAnthony |
| Soggetto topico |
Cayley graphs
Eigenvalues Cayley algebras |
| ISBN |
1-283-42780-X
9786613427809 0-19-987748-3 |
| Classificazione | MAT013000 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
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
2. Subgroups and quotients1. Coverings and quotients; 2. Subgroups and Schreier generators; Notes; Exercises; Student research project ideas; 3. The Alon-Boppana theorem; 1. Statement and consequences; 2. First proof: The Rayleigh-Ritz method; 3. Second proof: The trace method; Notes; Exercises; Student research project ideas; PART TWO: Combinatorial Techniques; 4. Diameters of Cayley graphs and expander families; 1. Expander families have logarithmic diameter; 2. Diameters of Cayley graphs; 3. Abelian groups never yield expander families: A combinatorial proof 4. Diameters of subgroups and quotients5. Solvable groups with bounded derived length; 6. Semidirect products and wreath products; 7. Cube-connected cycle graphs; Notes; Exercises; Student research project ideas; 5. Zig-zag products; 1. Definition of the zig-zag product; 2. Adjacency matrices and zig-zag products; 3. Eigenvalues of zig-zag products; 4. An actual expander family; 5. Zig-zag products and semidirect products; Notes; Exercises; Student research project ideas; PART THREE: Representation-Theoretic Techniques; 6. Representations of finite groups; 1. Representations of finite groups 2. Decomposing representations into irreducible representations3. 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 ?(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 basics2. 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 |
| Record Nr. | UNINA-9910778809803321 |
|
Krebs Mike
|
||
| Oxford ; ; New York, : Oxford University Press, c2011 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
The planar cubic Cayley graphs / / Agelos Georgakopoulos
| The planar cubic Cayley graphs / / Agelos Georgakopoulos |
| Autore | Georgakopoulos Agelos |
| Pubbl/distr/stampa | Providence, Rhode Island : , : American Mathematical Society, , 2017 |
| Descrizione fisica | 1 online resource (87 pages) : illustrations |
| Disciplina | 511/.5 |
| Collana | Memoirs of the American Mathematical Society |
| Soggetto topico |
Cayley graphs
Graph connectivity Graph theory |
| Soggetto genere / forma | Electronic books. |
| ISBN | 1-4704-4204-3 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910480876203321 |
|
Georgakopoulos Agelos
|
||
| Providence, Rhode Island : , : American Mathematical Society, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
The planar cubic Cayley graphs / / Agelos Georgakopoulos
| The planar cubic Cayley graphs / / Agelos Georgakopoulos |
| Autore | Georgakopoulos Agelos |
| Pubbl/distr/stampa | Providence, Rhode Island : , : American Mathematical Society, , 2017 |
| Descrizione fisica | 1 online resource (87 pages) : illustrations |
| Disciplina | 511/.5 |
| Collana | Memoirs of the American Mathematical Society |
| Soggetto topico |
Cayley graphs
Graph connectivity Graph theory |
| ISBN | 1-4704-4204-3 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introductory material and basic facts -- The finite and 1-ended cubic planar Cayley graphs -- The planar multi-ended Cayley graphs with 2 generators -- The planar multi-ended Cayley graphs generated by 3 involutions -- Outlook -- Bibliography. |
| Record Nr. | UNINA-9910795440903321 |
|
Georgakopoulos Agelos
|
||
| Providence, Rhode Island : , : American Mathematical Society, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
The planar cubic Cayley graphs / / Agelos Georgakopoulos
| The planar cubic Cayley graphs / / Agelos Georgakopoulos |
| Autore | Georgakopoulos Agelos |
| Pubbl/distr/stampa | Providence, Rhode Island : , : American Mathematical Society, , 2017 |
| Descrizione fisica | 1 online resource (87 pages) : illustrations |
| Disciplina | 511/.5 |
| Collana | Memoirs of the American Mathematical Society |
| Soggetto topico |
Cayley graphs
Graph connectivity Graph theory |
| ISBN | 1-4704-4204-3 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introductory material and basic facts -- The finite and 1-ended cubic planar Cayley graphs -- The planar multi-ended Cayley graphs with 2 generators -- The planar multi-ended Cayley graphs generated by 3 involutions -- Outlook -- Bibliography. |
| Record Nr. | UNINA-9910816743603321 |
|
Georgakopoulos Agelos
|
||
| Providence, Rhode Island : , : American Mathematical Society, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Soggetto topico
-
Cayley graphs
(5)
- Graph connectivity (3)
- Graph theory (3)
- Cayley algebras (2)
- Eigenvalues (2)