Fix any $k \geq 3$, and suppose I have a simple undirected graph $G=(V,E)$. I want a bound on the number of $k$ cycles in $G$ as a function of $|E|$. In particular, I would like to prove the following "conjectured" statement:

**W.T.S:** Given any $G=(V,E)$, the number of $k$-cycles in $G$ is at most $O(|E|^{k/2})$.

Clearly $\binom{|V|}{k}$ is tight if we only wanted a bound in terms of $|V|$, which comes from the complete graph on $|V|$ vertices. However, it is not as clear how to prove a tight upper bound in terms of $|E|$. The intuition for the conjecture is that, up to a constant, it seems that it should be optimal to arrange your edges in a complete graph (if $k$ is even, you do win a constant by taking a complete bipartite graph instead of a complete graph).

There seem to be a few proofs of this fact for triangles (k=3), for instance https://math.stackexchange.com/questions/823481/number-of-triangles-in-a-graph-based-on-number-of-edges. However, they don't generalize in any clear way to larger $k$. Moreover, I don't know of any k-cycle enumeration algorithms for k>3 that are parameterized by $|E|$ (the runtime of which would provide an upper bound).

For my application I actually only need bounds for odd $k$, but it seems that any such proof should generalize to any $k$. Also, to be clear, I'm only interested in asymtotic bounds here in terms of $|E|$ (getting tight constants seem much more difficult). Any suggestions or references that I missed would be much appreciated.