Friedman J. A proof of Alon's second eigenvalue conjecture and related problems. - Providence: American Mathematical Society, 2008. - vii, 100 p. - (Memoirs of the American Mathematical Society; Vol.195, N 910). - Bibliogr.: p.99-100. - ISBN 978-0-8218-4280-5; ISSN 0065-9266  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Chapter 1. Introduction ........................................ 1 Chapter 2. Problems with the Standard Trace Method ............. 7 1. The Trace Method .......................................... 7 2. Limitations of the Trace Expansion ....................... 12 Chapter 3. Background and Terminology ......................... 17 1. Graph Terminology ........................................ 17 2. Variable-Length Graphs and Subdivisions .................. 18 3. λ1 of a VLG .............................................. 19 4. Shannon's Algorithm and Formal Series .................... 19 5. Limiting Graphs .......................................... 22 6. Irreducible Eigenvalues .................................. 22 7. λ1 and Closed Walks for Infinite Graphs .................. 24 8. A Curious Theorem ........................................ 24 Chapter 4. Tangles ............................................ 27 Chapter 5. Walk Sums and New Types ............................ 33 1. Walk sums ................................................ 34 2. The Loop ................................................. 39 3. Forms, Types, and New Types .............................. 40 4. Motivation of Types and New Types ........................ 43 Chapter 6. The Selective Trace ................................ 47 1. The General Selective Trace .............................. 47 2. A Lemma on Selective Walks ............................... 47 3. Determining τfund for n,d ................................ 51 4. Determining τfund for n,d, n,d, and n,d ................. 51 Chapter 7. Ramanujan Functions ................................ 57 Chapter 8. An Expansion for Some Selective Traces ............. 59 Chapter 9. Selective Traces In Graphs With (Without) Tangles ............................................ 65 Chapter 10. Strongly Irreducible Traces ........................ 73 Chapter 11. A Sidestepping Lemma ............................... 77 Chapter 12. Magnification Theorems ............................. 81 Chapter 13. Finishing the n,d Proof ........................... 87