The spectral gap of random regular graphs

We bound the second eigenvalue of random d $$ -regular graphs, for a wide range degrees , using novel approach based on Fourier analysis. Let G n {G}_{n,d} be uniform graph vertices, and λ ( ) \lambda \left({G}_{n,d}\right) its largest by absolute value. For some constant c > 0 c>0 any degree with log 10 ≪ ≤ {\log}^{10}n\ll d\le cn we show that = 2 + o 1 − / \left({G}_{n,d}\right)=\left(2+o(1)\right)\sqrt{d\left(n-d\right)/n} asymptotically almost surely. Combined earlier results cover case sparse this fully determines asymptotic value all . To achieve this, introduce new methods use mechanisms from discrete analysis, combine them existing tools estimates graphs—especially those Liebenau Wormald.