Finding the connected subgraph of a given size with maximum number of edges, that includes a given vertex
It's NP-hard (by reduction from clique problem).
Let $G$ be an arbitrary graph, and we want to check if there exists a clique of size $m-1$ in this graph. This problem is NP-hard. By adding $v_1$ to the graph and connecting it to all vertices in $G$, we reduce the clique problem to your problem (since clique, when exists, has the maximum number of edges).