# What is the maximum possible degrees of a vertex of an MST

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?

• A minimum spanning tree could be a star, and so have a vertex of degree $n-1$, where $n$ is the number of vertices. – Yuval Filmus Feb 14 '18 at 12:26
• Ok, then at most n-1 – LSG Feb 14 '18 at 12:31
• @LSG Yes but that applies to any vertex of any $n$-vertex graph. – David Richerby Feb 14 '18 at 13:48

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 $n-1$.