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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?

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

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