Is the number of edges in the 3-connected graph with the minimum number of spanning trees always $\lceil {\frac{3}{2}}n\rceil$?
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1$\begingroup$ What are some examples of such 3c-min-span graphs? Presumably you know some classes that achieve the min... $\endgroup$– Joseph O'RourkeCommented Oct 14, 2010 at 19:40
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$\begingroup$ It seems there are no classes which achieve the minimum always. An example of a class which achieve a low number of spanning trees are Prism Graphs, but they do not always have the minimum number of spanning trees. $\endgroup$– utdiscantCommented Oct 14, 2010 at 20:50
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$\begingroup$ For what values of n do you know this to be true? $\endgroup$– j.c.Commented Oct 15, 2010 at 0:18
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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.)
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$\begingroup$ It is true that the 3-connected graph(s) with the minimum number of spanning trees must be found among the 3-connected graphs with the minimal number of edges, but this is not enough to draw a conclusion, since it is easy to find examples showing that minimal 3-connected graphs have more than ceil(3n/2) edges. For instance the Wheel Graph on 6 vertices is a minimal 3-connected graph, but has 10 edges. $\endgroup$ Commented Oct 15, 2010 at 1:57
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$\begingroup$ Also, I can't seem to find the paper ["3-connected graphs of minimal size".] which you are referring too, do you have a link? $\endgroup$ Commented Oct 15, 2010 at 2:01
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$\begingroup$ The actual title appears to be "P_3 - connected graphs of minimal size" in Ars Combinatorica. I found this by putting one of the author's names into my favorite search engine. This led me to the personal website of the author, which included a CV, which included a list of papers. (MathSciNet would have been even quicker.) $\endgroup$– JBLCommented Oct 15, 2010 at 2:31
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$\begingroup$ Yeah, that's a good point. Since the complex of not 3-connected graphs is homotopic to a wedge of spheres of the same dimension, I was under the impression that all of the minimal 3-connected graphs were of the same size - not the case. The critical graphs left over after an appropriately chosen discrete Morse matching are though... Do you have these graphs classified for small n? $\endgroup$ Commented Oct 15, 2010 at 5:00