Graph building with weighted nodes
I am not sure I understand the problem correctly, but let me try a definition:
Given a graph and an integer k.
Find the subset $S$ of $k+1$ vertices with max sum of weights such that $S$ induces a connected graph (i.e. there is a tree with k edges spanning $S$).

This sounds to me like the Steiner tree problem
http://en.wikipedia.org/wiki/Steiner_tree_problem

The steiner tree problem gives as input a graph, a set of terminals and asks for a set of edges from the graph (i.e. a tree) that connects all terminals.
The goal is to minimize the sum of the edge weights.

Gary and Johnson showed this NP-complete, I believe even for all edge weights 1.

Give the terminals a high weight such that any solution would need to take all terminals.
The rest of the vertices gets weight 1. Since a tree always has 1 edge less than number of nodes the question is find a set of nodes such that all terminals will be connected.

Hence your problem is NP-complete.
Much work has been done on approximation, there has been comparison of different implementations. I seem to remember there was a DIMACS challenge on the Steiner tree problem.