I need answers for part a, b, c, d Please provide complete explanation wherever required prove please prove how you get to your answer asap

5. (18 pts) All graphs discussed in this question have edge weights. MST denotes the minimum spanning tree (a) Let G! (Vi. E1) and G2 = (,Ea) be two graphs which are disconnected and for which MST's have i. Assume that, G (V, E) is modified by augmenting E by a unique edge joining Gi and G2. How ii. Assume now that, G = (V,E) is modified by augmenting E by two edges each joining G1 and G2. been computed. G-V, E) is a graph with V -Vi U V2 and E E1 U E2. does the MST for G look like? Prove your result! Discuss, possibly using case analysis, how the MST for G looks like? In particular, can both edges be part of the MST for G or not? Prove your result. (b) Let G (V, E) be a graph. Characterize when G has an MST and when not. (c) Let T be the minimum spanning tree for G. The deletion of the heaviest edge from G may not alter T. What can you say about the graph if the deletion of the heaviest graph edge does not alter T? Prove your result. What can you say otherwise? (d) Let G = (V,E) be a graph. The edge weights are 1, 2, 3, , E. Let T be an MST of G. What is the possible minimum weight of T? Does every MST of G have that minimum weight? Please, elaborate your answer (e) DT computes a shortest-path tree in a given graph G (V,E) using Dijkstra's algorithm. DT claims that this is the MST of G. Is DT right or wrong? Prove your answer. (f) DT also claims that a shortest path between two nodes of the given graph G = (V, E) is part of some MST of G. Is DT right or wrong? Prove your