co.combinatorics - Limit Distribution of Topological Form of Polyhedra with Large Number of Edges

Consider the set of all topologically inequivalent polyhedral graphs with $k$ edges, the number of which is given by Sloan sequence A002840 (1,0,1,2,2,4,12,22,58,158,448)).

Now define a topological form parameter as $beta:= (text{number of vertices}=v)/(k+2)$ and consider the distribution of the polyhedral graphs with $k$ edges as a function of $beta$. Due to duality the distribution is symmetric about $beta=1/2$. Due to the fact that for a planar graph $k le 3v-6$, the distribution vanishes outside the interval $beta in [1/3, 2/3]$.

Now a natural question is whether this distribution tends to a limiting distribution when the number of edges tends to infinity. Is it known whether such a limiting distribution exists - or will it be singular, i.e. concentrated with smaller and smaller variance around $beta=1/2$, as numerical data seem to suggest? Is there any nontrivial limit distribution theorem by means of rescaling?

EDIT: Some numerical data can be found under http://www.numericana.com/data/polycount.htm. Using these data gives the following values for the standard deviation of the distribution $p(beta)$ as a function of the number of edges: $sigma(k=21) = 0.029895922, sigma(k=27) = 0,027943943, sigma(k=33) = 0,02625827$.