Given an edge weighted graph, the maximum edge-weight connected graph (MECG) is a connected subgraph with a given number of edges and the maximal weight sum. Here we study a special case, i.e. the Constrained Maximum Edge-Weight Connected Graph problem (CMECG), which is an MECG whose candidate subgraphs must include a given set of k edges, then also called the k-CMECG. We formulate the k-CMECG into an integer linear programming model based on the network flow problem. The k-CMECG is proved to be NP-hard. For the special case 1-CMECG, we propose an exact algorithm and a heuristic algorithm respectively. We also propose a heuristic algorithm for the k-CMECG problem. Some simulations have been done to analyze the quality of these algorithms. Moreover, we show that the algorithm for 1-CMECG problem can lead to the solution of the general MECG problem.