  With graphs considered as natural models for many network design problems,
edge connectivity $\kappa'(G)$ and maximum number of edge-disjoint spanning
trees $\tau(G)$ of a graph $G$ have been used as measures for reliability and
strength in communication networks modeled as graph $G$ (see \cite{Cunn85,
Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72}
introduced the maximum subgraph edge connectivity $\overline{\kappa'}(G)=\max
\{\kappa'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their
applications in network design and by the established inequalities \[
\overline{\kappa'}(G)\ge \kappa'(G) \ge \tau(G), \] we present the following in
this paper:
  (i) For each integer $k>0$, a characterization for graphs $G$ with the
property that $\overline{\kappa'}(G) \le k$ but for any edge $e$ not in $G$,
$\overline{\kappa'}(G+e)\ge k+1$.
  (ii) For any integer $n > 0$, a characterization for graphs $G$ with $|V(G)|
= n$ such that $\kappa'(G) = \tau(G)$ with $|E(G)|$ minimized.
