For my homework I have a problem that I can't solve and it makes me wonder about 2 different MST:
Let $G=(V,E)$ be a graph that has a minimum spanning tree $T$.
I want to find another minimum spanning tree $T'$ that has at least 1 different edge $e'$ such that the weight of $e'$ is differ from any weight of edges in $T$.
If $T'$ doesn't exist I can claim that every 2 different MST must have the same weight for each edge.
My intuition says that this claim is wrong but on the other hand I can't find example of $T'$ to contradict this claim.