Given a connected, directed and weighted graph, Prim's algorithm may not necessarily generate the minimal spanning tree. Suppose we have such a graph $G$ with the special condition that for every pair of vertices $x, y \in G$ there exists a directed edge from $y \to x$ and a directed edge from $x \to y$ of equal weight. I was wondering if Prim's algorithm necessarily generates a minimal spanning tree if applied to $G$?
Further, suppose that we require a hamiltonian path of minimal weight between two vertices $x, y \in G$. Would the spanning tree generated by Prim's algorithm necessarily be less than or equal to the weight of this hamiltonian path?