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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.

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