I am working on the following exercise:
Is the following statements true or false?
Let $G=(V,E)$ be an undirected graph with edge weights $w_e$ for $e \in E(G$).
If $T^* = (V,E_{T^*})$ is a spanning tree such that $$max_{e \in E(T^*)} w_e \le max_{e \in E(T)} w_e$$ for all spanning trees $T=(V,E_{T})$. Does this imply that $\sum_{e \in E(T^*)} w_e \le \sum_{e \in E(T)} w_e$?
My guess is that this is true, but I can not prove it. Could you help me?
It is not true. Imagine $G$ to be pretty much any graph where one edge has weight 1 and the rest have weight 2. Then all spanning trees of $G$ will have maximum edge weight 2, but not all of them will be of minimum weight in total.