Number of edges in low complexity graphs Is the number of edges in the 3-connected graph with the minimum number of spanning trees always $\lceil {\frac{3}{2}}n\rceil$?
 A: Are the minimally 3-connected graphs (edge removal pushes you into a 2-connected graph) the same as the class of 3-connected graphs with the minimal number of spanning trees?  There is a nice classification of the minimal 3-connected graphs (or maximal not-3-connected graphs, I forget which direction he does it in) in Jonsson's book "Simplicial Complexes of Graphs".
I think there's also a classification of minimal 3-connected graphs inside a paper by David Fisher, Kathryn Fraughnaugh, and Larry Langley which could help ["3-connected graphs of minimal size".]  At the very least, they note that all 3-connected graphs on n vertices have at least 3/2(n-2) edges, and produce graphs achieving this bound.  All of your graphs with the minimal number of spanning trees must be one of these minimal 3-connected graphs (as if your hypothetical "minimal spanning tree" graph wasn't also a minimal 3-connected graph, removing edges to a minimal 3-connected graph would reduce the number of spanning trees.)
