Is the 3connected graph(s) on $n$ vertices with the minimum number of spanning trees always planar?
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Edit. As it turned out I was not using the right switch for plantri. This is therefore not an answer anymore but rather an extended comment for the case $n=11.$ As it turns out the minimal number of spanning trees of a 3connected planar graph of order 11 is 3965 and is attained by the graph on the figure bellow. As for the nonplanar 3connected graph I am yet to compute the answer. I'll post the result here as soon as it gets computed. 

