  We propose an efficient $\epsilon$-differentially private algorithm, that
given a simple {\em weighted} $n$-vertex, $m$-edge graph $G$ with a
\emph{maximum unweighted} degree $\Delta(G) \leq n-1$, outputs a synthetic
graph which approximates the spectrum with $\widetilde{O}(\min\{\Delta(G),
\sqrt{n}\})$ bound on the purely additive error. To the best of our knowledge,
this is the first $\epsilon$-differentially private algorithm with a
non-trivial additive error for approximating the spectrum of the graph. One of
the subroutines of our algorithm also precisely simulates the exponential
mechanism over a non-convex set, which could be of independent interest given
the recent interest in sampling from a {\em log-concave distribution} defined
over a convex set. Spectral approximation also allows us to approximate all
possible $(S,T)$-cuts, but it incurs an error that depends on the maximum
degree, $\Delta(G)$. We further show that using our sampler, we can also output
a synthetic graph that approximates the sizes of all $(S,T)$-cuts on $n$
vertices weighted graph $G$ with $m$ edges while preserving
$(\epsilon,\delta)$-differential privacy and an additive error of
$\widetilde{O}(\sqrt{mn}/\epsilon)$. We also give a matching lower bound (with
respect to all the parameters) on the private cut approximation for weighted
graphs. This removes the gap of $\sqrt{W_{\mathsf{avg}}}$ in the upper and
lower bound in Eli{\'a}{\v{s}}, Kapralov, Kulkarni, and Lee (SODA 2020), where
$W_{\mathsf{avg}}$ is the average edge weight.
