  Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be
the adjacency matrix of the random directed $d$-regular graph on $n$ vertices,
with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with
probability going to one as $n$ goes to infinity. The proof combines the method
of simple switchings and a recent result of the authors on delocalization of
eigenvectors of $A_n$.
