Super- is a more refined network reliability index than edge-connectivity. A graph is super- if every minimum edge-cut set is trivial (the set of edges incident at a node with the minimum degree ). This paper established the relation between diameter and super-: enlarging the order n under the given maximum degree and diameter D not only maximizes edge-connectivity, but also minimizes the number of minimum edge-cut sets, thus attaining super-. The following sufficient conditions for a digraph and graph G to be super - are derived. (1) Digraph G is super- if . (2) Graph G is super- if . These conditions are the best possible. From these, the de Bruijn digraph B(d, D), the Kautz digraph K(d, D), and most of the densest known graphs are shown to super-. Also, the digraph , which has been proposed as a maximally connected d-regular digraph with quasiminimal diameter (at most one larger than the lower bound) is proved to be super- for any and any order .
