Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e). For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace each edge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.

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Minimum Spanning Tree

A spanning tree is a sub-graph of a connected, undirected graph. It has all the vertices of a graph with the minimum possible number of edges. If a vertex is skipped, then it will not be a spanning tree. The edges of a spanning tree may or may not have weights.