Distribution of “topological form” for polyhedra with large number of edges

[andreasruedinger]

Consider the set of all topologically inequivalent polyhedra with k
edges (i.e. polyhedral graphs, number equal Sloan sequence A002840
(1,0,1,2,2,4,12,22,58,158,448)).

Define a form parameter as beta:= (number of vertices)/(number of
edges +2). Due to duality the distribution of beta is symmetric about
beta=1/2. Now a natural question is whether the distribution of beta
tends to a limiting distribution when the number of edges tends to
infinity. Do you know whether such a limiting distribution exists
(will it be singular, i.e. concentrated with smaller and smaller
variance around beta=1/2)? Is there any nontrivial limit theorem by
means of rescaling? Some numerical values can be found on
http://home.att.net/~numericana/data/polycount.htm suggesting that the
distribution concentrates around beta=1/2.