There have been many questions on this site about counting non-isormorphic graphs that are (1) strongly regular, (2) have a certain chromatic number, (3) connected, among many others.

What work has been done to count the number of non-isomorphic graphs that are connected and spanning that have exactly $n$ vertices and $m$ edges? Clearly, for $m < n-1$, this number is 0, and for $m = n-1$, this is the number of non-isomorphic spanning trees on $n$ vertices, which already does not seem to have a closed-form formula. However, this becomes even more complicated for larger values of $m$ (although there are some partial results for some graph classes with $m = n-1$ here).

If there is not a single expression (either recursive, or composed of a few "ingredients"), are there graph classes for which the number is known? This is helpful in determining an upper bound on the number of distinct reliability polynomials on graphs with $n$ vertices, $m$ edges.