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Alon's second eigenvalue conjecture : simplified and generalized Kohler, David-Emmanuel
Abstract
For a fixed graph, B, we study a probability model of random covering maps of degree n. Specifically, we study spectral properties of new eigenvalues of the adjacency matrix of a random covering, and its Hashimoto matrix (i.e., the adjacency matrix of the associated directed line graph). Our main theorem says that if B is d-regular, then for every positive epsilon, the probability that a random covering has a new adjacency eigenvalue greater than 2(d-1)^(1/2) + epsilon tends to zero as n tends to infinity. This matches the generalized Alon-Boppana lower bound. For general base graphs, B, Friedman conjectured in that the new eigenvalue bound holds with 2(d-1)^(1/2) replaced with the spectral radius of the universal cover of B. We refer to this conjecture as the generalized Alon conjecture; Alon stated this conjecture in the case where B has one vertex, i.e., the case of random d-regular graphs on n vertices. However, for some non-regular base graphs B, we cannot yet prove any non-trivial new eigenvalue upper bound with high probability. We use trace methods, as pioneered by Broder and Shamir for random, d-regular graphs; these methods were eventually refined by Friedman to prove the original Alon conjecture, i.e., in the case where B has one vertex. Our methods involve several significant simplifications of the methods of Friedman.
Item Metadata
Title |
Alon's second eigenvalue conjecture : simplified and generalized
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2013
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Description |
For a fixed graph, B, we study a probability model of random covering maps of degree n. Specifically, we study spectral properties of new eigenvalues of the adjacency matrix of a random covering, and its Hashimoto matrix (i.e., the adjacency matrix of the associated directed line graph).
Our main theorem says that if B is d-regular, then for every positive epsilon, the probability that a random covering has a new adjacency eigenvalue greater than 2(d-1)^(1/2) + epsilon tends to zero as n tends to infinity. This matches the generalized Alon-Boppana lower bound.
For general base graphs, B, Friedman conjectured in that the new eigenvalue bound holds with 2(d-1)^(1/2) replaced with the spectral radius of the universal cover of B. We refer to this conjecture as the generalized Alon conjecture; Alon stated this conjecture in the case where B has one vertex, i.e., the case of random d-regular graphs on n vertices. However, for some non-regular base graphs B, we cannot yet prove any non-trivial new eigenvalue upper bound with high probability.
We use trace methods, as pioneered by Broder and Shamir for random, d-regular graphs; these methods were eventually refined by Friedman to prove the original Alon conjecture, i.e., in the case where B has one vertex. Our methods involve several significant simplifications of the methods of Friedman.
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Genre | |
Type | |
Language |
eng
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Date Available |
2013-07-23
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-ShareAlike 3.0 Unported
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DOI |
10.14288/1.0073984
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2013-11
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-ShareAlike 3.0 Unported