Cycles of Specified Length in a Graph 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.)
An answer containing a recursive formula for the number of cycles can be found here.
I think the trace of $A^m$ gives the number of cycles of length $m$ of a graph $G$.