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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

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No, as a counterexample take a triangle with weights $4, 5, 6$ on edges. – Levon Haykazyan Jun 1 '12 at 18:56
Thanks but in your example, the path from any vertex that we choose to be the start vertex to one of the other 2 vertices is still the minimal between those 2 though. That's the property of triangle inequality, isn't it? – cody Jun 1 '12 at 19:07
I can't understand your comment. Consider the graph with vertices $a, b, c$ and edges $ab, bc, ca$ with assigned weights $4, 5, 6$. Then the spanning tree with minimal weight is $ab, bc$, but the shortest put from $a$ to $c$ is $ac$. Is this not correct? – Levon Haykazyan Jun 1 '12 at 19:14
My question is: if b is the first vertex we use to create the MST, is ba and bc the minimum cost path from b to any other vertex? The first vertex is important in my question because we may have different MST's if we start from different vertex. – cody Jun 1 '12 at 19:25
So this is an example showing that your assertion as stated is not correct, because you may be starting from the vertex $a$. – Levon Haykazyan Jun 1 '12 at 19:31

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:

MST example

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.

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