Abstract
AbstractWe bound the second eigenvalue of random ‐regular graphs, for a wide range of degrees , using a novel approach based on Fourier analysis. Let be a uniform random ‐regular graph on vertices, and be its second largest eigenvalue by absolute value. For some constant and any degree with , we show that asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of for all . To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on ‐regular random graphs—especially those of Liebenau and Wormald.
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