Does a closed walk necessarily contain a cycle? [HOMEWORK]
I asked my professor and he said that a counter example would be two nodes, by which the pathw ould go from one node and back. this would be a closed path but does not contain a cycle. But I am confused after looking at this again. Why is this not a cycle? Need there be another node?

I guess the answer depends on the exact definition of cycle. If it is as you wrote in your comment - a closed walk that starts and ends in the same vertex, and no vertex repeats on the walk (except for the start and end), then your example with two nodes is a cycle.
However, a definition of a cycle usually contains a condition of non-triviality stating that a cycle has at least three vertices. So a graph with two vertices is not a cycle according to this definition.
