The singularity probability of dense random regular graphs

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph on $n$ vertices, and let $A_n$ be its adjacency matrix. For every fixed $\lambda \in (0,1\mathrm{/}2)$, we prove that for any $\lambda (n-1)\le d\le (1-\lambda )(n-1)$, $\mathbb{P}(A_n\text{ is singular})\le e^{-cn}$, where $c>0$ depends only on $\lambda$, and $n$ is sufficiently large with $nd$ even.