  We study the adjacency matrices of random $d$-regular graphs with large but
fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon,
2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the
Green's functions can be approximated by those of certain infinite tree-like
(few cycles) graphs that depend only on the local structure of the original
graphs. This result implies that the Kesten--McKay law holds for the spectral
density down to the smallest scale and the complete delocalization of bulk
eigenvectors. Our method is based on estimating the Green's function of the
adjacency matrices and a resampling of the boundary edges of large balls in the
graphs.
