# Asymptotic Distribution of Form of Polyhedra

 Importance: Medium ✭✭
 Author(s): Rüdinger, Andreas
 Subject: Graph Theory » Basic Graph Theory
 Keywords: polyhedral graphs, distribution
 Posted by: andreasruedinger on: May 9th, 2009
Problem   Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?

Consider the set of all topologically inequivalent polyhedra on a sphere with k edges (i.e. polyhedral graphs, Sloan Sequence A002840 ). Due to duality the distribution of the form parameter is symmetric about . 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 Counting Polyhedra suggesting that the distribution concentrates around .