Proof of Alon’s Second Eigenvalue Conjecture and Related Problems.
- 1st ed.
- 1 online resource (114 pages)
- Memoirs of the American Mathematical Society ; v.195 .
- Memoirs of the American Mathematical Society .
Intro -- Contents -- Chapter 1. Introduction -- Chapter 2. Problems with the Standard Trace Method -- 1. The Trace Method -- 2. Limitations of the Trace Expansion -- Chapter 3. Background and Terminology -- 1. Graph Terminology -- 2. Variable-Length Graphs and Subdivisions -- 3. λ[sub(1)] of a VLG -- 4. Shannon's Algorithm and Formal Series -- 5. Limiting Graphs -- 6. Irreducible Eigenvalues -- 7. λ[sub(1)] and Closed Walks for Infinite Graphs -- 8. A Curious Theorem -- Chapter 4. Tangles -- Chapter 5. Walk Sums and New Types -- 1. Walk sums -- 2. The Loop -- 3. Forms, Types, and New Types -- 4. Motivation of Types and New Types -- Chapter 6. The Selective Trace -- 1. The General Selective Trace -- 2. A Lemma on Selective Walks -- 3. Determining τ[sub(fund)] for G[sub(n,d)] -- 4. Determining τ[sub(fund)] for H[sub(n,d)], I[sub(n,d)], and J[sub(n,d)] -- Chapter 7. Ramanujan Functions -- Chapter 8. An Expansion for Some Selective Traces -- Chapter 9. Selective Traces In Graphs With (Without) Tangles -- Chapter 10. Strongly Irreducible Traces -- Chapter 11. A Sidestepping Lemma -- Chapter 12. Magnification Theorems -- Chapter 13. Finishing the G[sub(n,d)] Proof -- Chapter 14. Finishing the Proofs of the Main Theorems -- Chapter 15. Closing Remarks -- Glossary -- Bibliography.