Gutman index‎, ‎edge-Wiener index and edge-connectivity

‎We study the Gutman index ${\rm Gut}(G)$ and the edge-Wiener index $W_e (G)$ of connected graphs $G$ of given order $n$ and edge-connectivity $\lambda$‎. ‎We show that the bound ${\rm Gut}(G) \le \frac{2^4 \cdot 3}{5^5 (\lambda+1)} n^5‎ + ‎O(n^4)$ is asymptotically tight for $\lambda \ge 8$‎. ‎We improve this result considerably for $\lambda \le 7$ by presenting asymptotically tight upper bounds on ${\rm Gut}(G)$ and $W_e (G)$ for $2 \le \lambda \le 7$‎.