The Problem of n-connection subgraph discovery (n-CSDP for short) is to find a small sized subgraph that can well capture the relationship among the n given nodes in a large graph. However there have been very few researches directly addressing the CSDP problem. Furthermore the currently available methods, for example, the electricity analogues based algorithm can only be suitable for tackling the 2-Keynodes CSDP and does not work anymore when n is greater than two. To deal with this problem, we propose an effective approach to discover the subgraph in two stages. In the first stage we propose a neighbor-growth based method to extract a relatively bigger candidate subgraph compared with that of result subgraph. In the second stage an evolutionary algorithm for optimizing the result subgraph is proposed. For this purpose, UTM code, a transformed representation of the adjacent matrix of graphs is designed to encode the topology of subgraph as individuals. Then corresponding evolutionary operators able to be directly performed on UTM code are given. Thus the efficiency of the algorithm is largely improved. The experimental results obtained on two real large scale graphs with different topology characteristics demonstrate that our method solves n-connection subgraph discovery problems effectively.