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Asymptotic Distribution of Form of Polyhedra

Importance: Medium ✭✭
Author(s): Rüdinger, Andreas
Subject: Graph Theory
» Basic Graph Theory
Keywords: polyhedral graphs, distribution
Recomm. for undergrads: no
Posted by: andreasruedinger
on: May 9th, 2009
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 $?

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 $ \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 Counting Polyhedra suggesting that the distribution concentrates around $ \beta=1/2 $.

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