Remarks: All the graphs here are without self loops and parallel edges, and anti$-$parallel edges. In all the algorithms, always explain their correct ness and analyze their time complexity. The time complexity should be as small as possible. A correct algorithm with large time complexity may not get full credit.

$$ Question $5$: Given an undirected graph with costs on the edges that are pairwise distinct $($no cost appears twice$).$ Show that there is a unique minimum spanning tree.

Remark: Saying that the Prim algorithm returns a unique tree or that the Kruskal algorithm returns a unique tree is not an answer.