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 $2 le lambda le 7$‎.