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

Abstract

A d-regular graph has largest or first (adjacency matrix) eigenvalue λ 1 = d. Consider for an even d ≥ 4, a random d-regular graph model formed from d/2 uniform, independent permutations on {1, . . . , n}. We shall show that for any ǫ > 0 we have all eigenvalues aside fromWe also show that this probability is at most 1 − c/n τ ′ , for a constant c and a τ ′ that is either τ or τ + 1 ("more often" τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd.These theorems resolve the conjecture of Alon, that says that for any ǫ > 0 and d, the second largest eigenvalue of "most" random dregular graphs are at most 2 √ d − 1 +ǫ (Alon did not specify precisely what "most" should mean or what model of random graph one should take).

Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

“…In this sense, Erdős-Rényi graphs asymptotically satisfy the graph Riemann hypothesis, itself is a plausible extension of the notion of Ramanujan graphs to the non-regular case. This may be seen as an analogue of Friedman's Theorem [9] in the context of Erdős-Rényi graphs. Similarly, for the Stochastic Block Model, our main result is analogue to recent results on the eigenvalues of random n-lifts of base graphs, see [10,6].…”

“…Hence Ramanujan graphs are regular graphs with maximal spectral gap between the first and second eigenvalue moduli. A celebrated result of Friedman [9] states that random k-regular graphs achieve this lower bound with high probability as their number of nodes n goes to infinity.…”

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“…Vertices of the wrong degree are essentially the only obstacle to approximating the complete graph by a random graph. Friedman [Fri08] proved that if we choose a random graph in which every vertex has degree d, then it is probably very close to being a Ramanujan graph.…”

Learning Markov graphs up to edit distance

“…We choose d perfect matchings on p nodes, each perfect matching chosen uniformly at random, and form a multigraph resulting from the union of the edges in the matchings. We refer to the set of all such multigraphs on p nodes, constructed in this fashion, as H. The uniform distribution over the choices of perfect matchings also defines a probability distribution over H. We have the following lemma for this distribution [21]:…”