Counting the Number of Simple Paths in Undirected Graph How can I go about determining the number of unique simple paths within an undirected graph? Either for a certain length, or a range of acceptable lengths.

Recall that a simple path is a path with no cycles, so I'm talking about counting the number of paths with no cycle.
There are several algorithms that count the simple paths of length $k$ in $f(k)n^{k/2+O(1)}$ time, which is a whole lot better than brute force ($O(n^k)$ time). See e.g. Vassilevska and Williams, 2009.