Asymptotic Distribution of Form of Polyhedra

Importance: Medium ✭✭
Author(s): Rüdinger, Andreas
Recomm. for undergrads: no
Posted by: andreasruedinger
on: May 9th, 2009

\begin{problem} Consider the set of all topologically inequivalent polyhedra with $k$ edges. Define a form parameter for a polyhedron as $\beta:= v/(k+2)$ where $v$ is the number of vertices. What is the distribution of $\beta$ for $k \to \infty$? \end{problem}

Consider the set of all topologically inequivalent polyhedra on a sphere with k edges (i.e. polyhedral graphs, \href [Sloan Sequence A002840]{http://www.research.att.com/~njas/sequences/A002840} ). Due to duality the distribution of the form parameter $\beta:= v/(k+2)$ 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. Is there any nontrivial limit theorem by means of rescaling? Some numerical values can be found on \href[Counting Polyhedra]{http://home.att.net/~numericana/data/polycount.htm} suggesting that the distribution concentrates around $\beta=1/2$.