Spectrum of random d‐regular graphs up to the edge

Consider the normalized adjacency matrices of random d ‐regular graphs on N vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N , that is, . (ii) All eigenvectors of random d ‐regular graphs are completely delocalized.