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Prove - The spanning tree $M$ is a minimum spanning tree (MST) for $G$ with weights $c > 0$ if and only if $M$ is an MST for $G$ with weights $c^x$ for $x > 0$.

I am confused, how to prove this, as $c$ could be between $0$ and $1$, and then multiplying edge weights by $c$, will decrease weight. How can I prove the proposition?

John L.
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G K
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    I think the key property is that the relationship between edge weights, e.g. c1 < c2 is preserved after raising to positive powers. This carries through to sums of edge weights. – Matthew C Apr 30 '23 at 16:05
  • if the edge weight is c=0.5 raising 0.5^2 will decrease weight, but is the edge weight is c= 1.5, raising 1.5^2 will increase its weight. if the graph G has different weights, raising them to powers can change MST quite a lot. – G K Apr 30 '23 at 19:52
  • Right, but when you are trying to find a minimum subset of edges, you care about the relationship between edges (rather than how an edge is weighted under different values of x; you should treat x as if it's a fixed positive value throughout your proof) – Matthew C Apr 30 '23 at 20:00

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