Minimum spanning tree using DFS and BFS
Briefly, the answer is no, we cannot construct minimum spanning tree for an un-directed graph with distinct weights using BFS or DFS algorithm. This post provides a counterexample. 

Computing MST using DFS/BFS would mean it is solved in linear time, but (as Yuval Filmus commented) it is unknown if such algorithm exists. However, there is an expected-linear-time randomized algorithm computing the MST.