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Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any ǫ > 0 " most " graphs have their largest non-trivial eigenvalue at most 2 √ d − 1+ ǫ in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 √ d − 1 then the graph is said to be Ramanujan. These graphs have important… (More)

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