Results on the edge-connectivity of graphs

It is shown that if G is a graph of order p >= 2 such that deg u + deg v >= p - 1 for all pairs u, v of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg v >= p for all pairs u, v of nonadjacent vertices, then either p is even and G is isomorphic to K"p"2 x K"2 or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.