In a graph, if I expand a vertex to a minimum spanning tree, does this entail that the path(s) obtained by walking from the start vertex to any other vertex along the tree are minimal? Thanks
In general, one should not expect an MST to provide the shortest paths, even from a vertex used first in the construction of the MST. In addition to the counterexample of a triangle with weights 4,5,6 on edges, given by Levon Haykazyan, consider this example from Wikipedia:
The nearly-horizontal edge with weight $9$ will not be included in any MST, because the edge with weight $8$ is the better way to connect the cluster on the left to the rest of the graph. Traveling along the MST between the endpoints of this weight 9 edge takes at least 10 units.