Consider some graph that has 10 edges and 16 vertices.
What is the maximum possible length of a longest path in this graph?The maximum is obviously 10, because there are 10 edges that can be arranged in a path.
What is the minimum possible length of a longest path in this graph?
The minimum possible longest path can't be 0, because we have a non-zero number of edges. A graph with edges from the first 10 vertices to the 11th has only length 1 paths. Hence, the minimum longest path is 1.