| Abstract: | Let G be a graph and let k′(G) be the edge-connectivity of G. The strength of G, denoted by k?′(G), is the maximum value of k′(H), where H runs over all subgraphs of G. A simple graph G is called k-maximal if k?′(G) ≤ k but for any edge e ∈ E(Gc), k?′(G + e) ≥ k + 1. Let G be a k-maximal graph of order n. In 3], Mader proved |E(G)| ≤ (n - k)k + ( ). In this note, we shall show (n - 1)k - ( ) In?n/k + 2)? ≤ |E(G|, and characterize the extremal graphs. We shall also give a characterization of all k-maximal graphs. |
