  We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic
of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the
rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$
stabilizes for large $n$. That means, if $n \geq g \geq 2$, then there are
isomorphisms $H^k(\mathcal{MG}_{g,n};\mathbb{Q})^{{\mathbb S}_n} \rightarrow
H^k(\mathcal{MG}_{g,n+1};\mathbb{Q})^{{\mathbb S}_{n+1}}$ for all $k$.
