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.