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A proof of Alon’s second eigenvalue conjecture and related problems

About this Title

Joel Friedman

Publication: Memoirs of the American Mathematical Society
Publication Year 2008: Volume 195, Number 910
ISBNs: 978-0-8218-4280-5 (print); 978-1-4704-0516-8 (online)
MathSciNet review: 2437174
MSC: Primary 05C50; Secondary 05C80, 05C81, 68R10

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Table of Contents


  • 1. Introduction
  • 2. Problems with the standard trace method
  • 3. Background and terminology
  • 4. Tangles
  • 5. Walk sums and new types
  • 6. The selective trace
  • 7. Ramanujan functions
  • 8. An expansion for some selective traces
  • 9. Selective traces in graphs with (without) tangles
  • 10. Strongly irreducible traces
  • 11. A sidestepping lemma
  • 12. Magnification theorems
  • 13. Finishing the $\mathcal {G}_{n,d}$ proof
  • 14. Finishing the proofs of the main theorems
  • 15. Closing remarks