This question is based on the cut lemma for minimum spanning trees. Let G = (V, E) be a connected graph where each edge has a positive weight. If for any cut of G, there is a unique edge in the cut of minimum weight then show that minimum spanning tree of G is unique. Show that the converse may not be true using an example. I.e., construct a graph G that has a unique minimum spanning tree, but there are cut(s) in G containing multiple edges having the minimum weight.