Dear Gaurish, could you precise? Your question leads to disambiguation. Is your graph acyclic? I hope yes, since I understand your question as follows:
CONVERT = keep all vertices and edges, and choose a root. There are more possibilities, from which you want that (those) where the height of the tree is minimal.
You can connect all the nodes at minimum cost with Kruskal's or Prim's algorithm. On the other hand, if you need to minimize the distance from a particular node, you can use Dijkstra's algorithm.