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A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems
 
Joel Friedman University of British Columbia, Vancouver, BC, Canada
Front Cover for A Proof of Alon's Second Eigenvalue Conjecture and Related Problems
Available Formats:
Electronic ISBN: 978-1-4704-0516-8
Product Code: MEMO/195/910.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
Front Cover for A Proof of Alon's Second Eigenvalue Conjecture and Related Problems
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  • Front Cover for A Proof of Alon's Second Eigenvalue Conjecture and Related Problems
  • Back Cover for A Proof of Alon's Second Eigenvalue Conjecture and Related Problems
A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems
Joel Friedman University of British Columbia, Vancouver, BC, Canada
Available Formats:
Electronic ISBN:  978-1-4704-0516-8
Product Code:  MEMO/195/910.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1952008; 100 pp
    MSC: Primary 68; 05;

    A \(d\)-regular graph has largest or first (adjacency matrix) eigenvalue \(\lambda_1=d\). Consider for an even \(d\ge 4\), a random \(d\)-regular graph model formed from \(d/2\) uniform, independent permutations on \(\{1,\ldots,n\}\). The author shows that for any \(\epsilon>0\) all eigenvalues aside from \(\lambda_1=d\) are bounded by \(2\sqrt{d-1}\;+\epsilon\) with probability \(1-O(n^{-\tau})\), where \(\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1\). He also shows that this probability is at most \(1-c/n^{\tau'}\), for a constant \(c\) and a \(\tau'\) that is either \(\tau\) or \(\tau+1\) (“more often” \(\tau\) than \(\tau+1\)). He proves related theorems for other models of random graphs, including models with \(d\) odd.

  • Table of Contents
     
     
    • Chapters
    • 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
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1952008; 100 pp
MSC: Primary 68; 05;

A \(d\)-regular graph has largest or first (adjacency matrix) eigenvalue \(\lambda_1=d\). Consider for an even \(d\ge 4\), a random \(d\)-regular graph model formed from \(d/2\) uniform, independent permutations on \(\{1,\ldots,n\}\). The author shows that for any \(\epsilon>0\) all eigenvalues aside from \(\lambda_1=d\) are bounded by \(2\sqrt{d-1}\;+\epsilon\) with probability \(1-O(n^{-\tau})\), where \(\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1\). He also shows that this probability is at most \(1-c/n^{\tau'}\), for a constant \(c\) and a \(\tau'\) that is either \(\tau\) or \(\tau+1\) (“more often” \(\tau\) than \(\tau+1\)). He proves related theorems for other models of random graphs, including models with \(d\) odd.

  • Chapters
  • 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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