# Uniqueness of minimum spanning tree

If G has a unique minimum spanning tree, does that mean the edge weights in G are also unique? if yes why and if no why?

• What did you try? Where did you get stuck? – David Richerby May 16 '19 at 16:22
• I know that if the edge weights in G are unique, then G has a unique minimum spanning tree I'm just curious what we reverse the condition. – JKLM May 16 '19 at 16:23
• Trees have a unique minimum spanning tree. – David Richerby May 16 '19 at 16:24
• but that did not answer my question :) – JKLM May 16 '19 at 16:25
• If you think about it a bit more, it does. Or, rather, it contains exactly enough information for you to answer your question. – David Richerby May 16 '19 at 16:29

If $$G$$ is a tree, it has a unique MST whatever its weights are. The weights could be unique, all the same, anything.
The question is: If $$G$$ has a unique MST, then are the edge weights necessarily distinct?
The answer is no. If $$G$$ is a tree with all edges having the same weight, then $$G$$ has a unique MST.