How many minimal $k$-cycles can you put in a graph? I wish to construct an unweighted, undirected graph on $n$ nodes that maximizes the number of minimal cycles of size $\ge k$.  What is known about this problem?  How many cycles can I squeeze in?

Additionally, how does the problem change if I require that my graph has $\ge m$ edges?

Clarification: I care purely about the existence of a many-cycled graph; it doesn't need to be efficiently constructable.
If $k$ is divisible by 2 we can take a clique on $n/2$ vertices and subdivide each edge once.
This graph will have ${n/2 \choose k/2} \cdot (k/2-1)!/2$ cycles of length $k$.

Hence the total number of cycles is $\frac{1}{2} \sum_{i=k/2}^{i=n/2} {n/2 \choose i} \cdot (i-1)!$