Let $G=(V,E)$ be a graph, and $A$ be its adjacency matrix. Define $n = |V|$.
Given $A$ and a natural number $m \le n$, I'm interested in the following problem:
How many simple cycles of length $m$ exist in $G$?
By simple cycle, I mean no repeated vertices along the cycle is allowed (other than the starting and ending vertices, which coincide).
The problem is NP-hard. However, I'm not asking its complexity; I'm merely interested in whether there is a closed-form expression for computing it. (Thus, computing the expression can be NP-hard.)