Finding number of cycles of length $k$ in a graph
When $k$ is part of the input, the problem of deciding if $G$ contains a simple cycle of length $k$ is NP-complete. For every fixed $k$, the problem can be solved in either $O(VE)$ time, or $O(V^\omega \log V)$ time. Flum and Grohe [1] showed that counting cycles and paths of length $k$ in both directed and undirected graphs, parameterized by $k$, is #W[1]-complete. 

For $3 \leq k \leq 7$, one can count the $k$-cycles in $O(V^\omega)$ time, where $\omega &lt; 2.376$ is the exponent of matrix multiplication. This is the result of Alon, Yuster and Zwick [2]. The paper also contains methods for finding simple cycles of length exactly $k$, where $k \geq 3$.



[1] Flum, Jörg, and Martin Grohe. "The parameterized complexity of counting problems." SIAM Journal on Computing 33.4 (2004): 892-922.

[2] Alon, Noga, Raphael Yuster, and Uri Zwick. "Finding and counting given length cycles." Algorithmica 17.3 (1997): 209-223.