The graph shown below has 8 edges with distinct integer edge weights. The…

Key idea: each edge not in the given minimum spanning tree must have weight strictly greater than the largest edge on the unique path between its endpoints in the MST (so the given MST remains minimum). All edge weights are distinct integers.

• Identify the three edges not in the MST: these are the edges joining A–B, C–D, and E–D.

• For A–B: the path in the MST is A–C–B with edge weights 9 and 2, so the maximum on the path is 9. Therefore A–B must be an integer > 9. The smallest unused integer > 9 is 10, so assign A–B = 10.

• For E–D: the path in the MST is E–F–D with edge weights 4 and 6, so the maximum on the path is 6. Therefore E–D must be an integer > 6. The smallest available is 7, so assign E–D = 7.

• For C–D: the path in the MST is C–B–E–F–D with edge weights 2, 15, 4, 6, so the maximum is 15. Therefore C–D must be an integer > 15. The smallest available is 16, so assign C–D = 16.

Compute the total:

• Sum of given MST edges = 36 (9 + 2 + 15 + 4 + 6).

• Sum of the three non-MST edges chosen = 10 + 7 + 16 = 33.

• Total minimum possible sum of all 8 edges = 36 + 33 = 69.

Therefore the minimum possible total weight of all 8 edges is 69.