  We consider the statistics of extreme eigenvalues of random $d$-regular
graphs, with $N^{\mathfrak c}\leq d\leq N^{1/3-{\mathfrak c}}$ for arbitrarily
small ${\mathfrak c}>0$. We prove that in this regime, the fluctuations of
extreme eigenvalues are given by the Tracy-Widom distribution. As a
consequence, about 69% of $d$-regular graphs have all nontrivial eigenvalues
bounded in absolute value by $2\sqrt{d-1}$.
