What is the number that a minimum spanning tree can have a vertex with degree at most? Is there any rule? Is it related to the number of vertex or edge? Or not?
Suppose that the graph is already a tree. Then the minimum spanning tree is the graph itself. In particular, if we take a star on $n$ vertices, we obtain a minimum spanning tree having a vertex of degree $n-1$. This is optimal, since in a graph containing $n$ vertices, all degrees are at most $n-1$.