So like imagine a pentagon for example, so each vertex is adjacent to the next vertex, and then each of those vertices shares an edge with one node in the middle. So there are 6 vertices here, $n = 6$. How many total cycles are in this graph? And then how would you generalize that to any graph with $n \geq 4$ vertices?
I found that for the pentagon example as described, there are $13$ cycles (if I calculated correctly). There are $5$ cycles of length $3 (n-1), 4$ cycles of length $2 (n-2)$, and $4$ cycles of length $5$ ($n-3+1$, the extra $+1$ is for the entire pentagon being a cycle of length $5$). So the formula for this would be $n-1+n-2+n-3+1 = n+n-2+n-3.$
Similarly, for a graph of this description with $n=4$, so a triangle with a node in the middle, there are $7$ cycles. There are $4$ cycles of length $3$ and $3$ cycles of length $4.$ So the formula for this would be $n+n-1.$
I am having a hard time generalizing these into formulas for just a graph with $n$ vertices in general (with an answer in closed form).