Let G be a simple connected graph on 2n vertices with perfect matching. For a given positive integer k (0≤k≤n−1), G is k-extendable if any matching of size k in G is contained in a perfect matching of G. It is proved that if G is a k-extendable graph on 2n vertices with k≤n/2, then either G is bipartite or the connectivity of G is at least 2k. As a corollary, we show that if G is a maximal k-extendable graph on 2n vertices with n+2≤2k+1, then G is Kn; n if k+16≤n and G is K2n if 2k+16≤2n−1. Moreover, if G is a minimal k-extendable graph on 2n vertices with n+1≤2k+1 and k+16≤n thenthe minimum degree of G is k+1. We also discuss the relationship between the k-extendable graphs and the Hamiltonian graphs.