Super-$\lambda$ is a more refined network reliability index than edge-connectivity; $G$ is super-$\lambda$ if every minimum edge-cut set is trivial (the set of edges incident at a node with the minimum degree $\delta$). This paper clarifies the relations between diameter and super-$\lambda$; enlarging the order (the number of nodes) $n$ under the given maximum degree $\Delta$ and a diameter $D$ results in not only maximizing edge-connectivity but also minimizing the number of minimum edge-cut sets, thus attaining super-$\lambda$. The following sufficient conditions for digraph and graph $G$ to be super-$\lambda$ are derived. Digraph $G$ is super-$\lambda$ if $n greater than \delta (\frac{\Delta^{D-1}-1}{\Delta -1} + 1) + \Delta^{D-1}$ Digraph $G$ is super-$\lambda$ if $n greater than \delta (\frac{(\Delta-1)^{D-1}-1}{\Delta -2} + 1) + (\Delta-1)^{D-1}$ These conditions are best possible. From these, de Bruijn digraph $B(d,D)$, Kautz digraph $K(d,D)$, and most of the densest known graphs (listed in [3,9] are shown to be super-$\lambda$. Also, the digraph $G^{\ast}_{B}(n,d)$ proposed in [24] as a maximally connected $d$-regular digraph with quasiminimal diameter (at most one larger than the lower bound) is proved to be super-$\lambda$ for any $d$ greater than 2 and any order $n$ greater than $d^{3}$.