Find a graph path whose least edge weight is as large as possible. In a connected undirected graph, use the unique route in a maximum spanning tree. Deleting a least-weight edge on that route gives a separating cut of a graph; a heavier crossing edge would improve the tree's total weight, so every competing route contains an edge no heavier than the deleted one. This proves the bottleneck optimum without assuming distinct weights.
Articles by others on the same topic
The Widest Path Problem is a problem in graph theory that involves finding a path between two vertices in a weighted graph such that the minimum weight (or capacity) of the edges along the path is maximized. In other words, instead of minimizing the cost or distance as in traditional shortest path problems, the goal is to maximize the "widest" or largest bottleneck along the path between two nodes.