(a) Let G = (V, E) be a loop-free weighted connected undirected graph. If e1 ∈ E with wt(ei) < wt(e) for all other edges e1 ∈ E, prove that edge e1 is part of every minimal spanning tree for G.
(b) With G as in part (a), suppose that there are edges e1, e2 ∈ E with wt(e1) < wt(e2) < wt(e) for all other edges e ∈ E. Prove or disprove: Edge e2 is part of every minimal spanning tree for G.