  Consider the normalized adjacency matrices of random $d$-regular graphs on
$N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as
$\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We
prove that the optimal (up to an extra $N^{{\rm o}_N(1)}$ factor, where ${\rm
o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely,
denote $\gamma_i$ as the classical location of the $i$-th eigenvalue under the
Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\rm
o}_N(1)}$,
  \begin{align*}
  |\lambda_i-\gamma_i|\leq \frac{N^{{\rm o}_N(1)}}{N^{2/3}
(\min\{i,N-i+1\})^{1/3}},\quad \text{ for all } i\in \{2,3,\cdots,N\}.
  \end{align*}
  In particular, the fluctuations of extreme eigenvalues are bounded by
$N^{-2/3+{\rm o}_N(1)}$. This gives the same order of fluctuation as for the
eigenvalues of matrices from the Gaussian Orthogonal Ensemble.
