The maximum value of 𝑥 such that the edge between the nodes B and C is…

Answer: 5

Reasoning:

• Key fact: An edge appears in every minimum spanning tree if there exists a cut separating its endpoints for which that edge is the unique minimum-weight edge crossing the cut.

• Consider cuts that separate node B and node C. Evaluate each cut and the crossing edge weights:

• Cut with set {B}: crossing edges are AB = 7, BD = 3, and BC = x. For BC to be the unique minimum here requires x < 3, so x ≤ 2 (integer).

• Cut with set {B, D}: crossing edges are AB = 7, AD = 6, DC = 8, and BC = x. For BC to be the unique minimum here requires x < 6, so x ≤ 5 (integer).

• Cuts {B, A} and {B, A, D} have AC = 1 crossing, so BC cannot be the unique minimum for those cuts unless x < 1, which is not relevant for maximizing x.

• Because we only need one cut where BC is the unique minimum, the best (largest) integer upper bound comes from the cut {B, D}, giving x ≤ 5.

• Therefore, the maximum integer value of x that guarantees the edge between B and C is included in every minimum spanning tree is 5.