Depth of edge ideals and vertex connectivity of finite graphs

Let $G$ be a finite graph on $[n]:=\{1,\dots ,n\}$ and $\kappa (G)$ its vertex connectivity. Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}S\mathrm{/}I(G^c)\le \kappa (G)+1$. We give a sharp lower bound of $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}S\mathrm{/}I(G^c)$ in terms of $n$ and $\kappa (G)$. Furthermore, a sharp lower bound of $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}S\mathrm{/}I(G^c{)}^2$ as well as that of $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}S\mathrm{/}I(G^c{)}^{(2)}$ in terms of $n$ and $\kappa (G)$ is given.