total cycles in a cycle graph, $C_n$

Question

I'm wondering how many cycles are in any cycle graph $C_n$. I am thinking that any of the n vertices could initiate a cycle and then we could also go the reverse direction for each of the cycles. I'm quite new to graph theory. But I'm also not sure if these are all the same cycle, and any cycle graph, $C_n$, would just have 1 total cycle.

Answer 1

I think this depends on your definition of a cycle.

If your definition is a connected subgraph such that all vertices have degree 2, then there is only one cycle.

However, if you define a cycle to be $v_0 v_1 \dots v_k v_0$ then I guess you could make a case that there are $n$ such cycles.

Edit: as per the comment, we should stick to the canonical definition of a cycle.