Let $G$ be a simple graph and denote by $\tau(G)$ the number of spanning trees of $G$.
There are many results related to $\tau(G)$ for certain types of graphs. For example one of the prettiest (to me) is that if $C_n^2$ denotes the square of a cycle, then
$$\tau(G) = n^2 f_n$$
where $f_n$ is the $n$'th Fibonacci number.
What I am wondering is if there are any known applications (practical and theoretical) of knowing $\tau(G)$ for certain graphs?