Consider a weighted undirected graph with positive edge weights and let (u, v)…

Conclusion: weight(u, v) ≥ 12.

Reason:

• By the triangle inequality for shortest paths, the shortest distance from the source to v is at most the shortest distance to u plus the edge weight: 65 ≤ 53 + weight(u, v).

• Rearranging gives weight(u, v) ≥ 65 − 53 = 12.

• Thus the only statement that is always true is that the edge weight is at least 12. The edge weight could equal 12 (so statements claiming strictly greater than 12 are false) and it could be larger than 12 if the direct edge is not used by the shortest path to v (so statements claiming ≤ 12 are false).

• Example to illustrate equality: if the shortest path to u is 53 and the edge (u, v) has weight 12 and that edge is used on the shortest path to v, then the shortest path to v is 53 + 12 = 65, so weight(u, v) = 12 is possible.