It's a significant problem to find the densest subgraph in many research areas. Now, there are so many groups in the WeChat, QQ and other online chat softwares. In order to find the closely connected subgraphs with the maximum average density for the vertex-weighted graph, we introduce the concept of the densest subgraph, and then an exactly algorithm is presented as an extension of the Goldberg’s algorithm. By theoretical analysis, we prove that our proposed algorithm is correct and it runs in polynomial time. Thus, our proposed algorithm can be used to find the closely connected subgroups with the maximum average density in fact.