Depth stability of edge ideals

Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $depth S/I_G^k$ stabilizes is denoted by $dstab(I_G)$. We show that $dstab(I_G)<ell(I_G)$ where $ell(I_G)$ denotes the analytic spread of $I$. For trees we give a stronger upper bound for $dstab(I_G)$. We also show for any two integers $1leq a<b$ there exists a tree for which $dstab(I_G)=a$ and $ell(I_G)=b$.