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ArticleOriginal scientific text

Title

On rational radii coin representations of the wheel graph

Authors 1, 2

Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA
  2. Mathematics Department, Hood College, Frederick, MD, USA

Abstract

A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the cosinus of the internal angles formed by the center coin and two consecutive petals of the flower. In that process we also derive a recursion that these irreducible polynomials satisfy.

Keywords

ENG
planar graph, coin graph, flower, polynomial ring, Galois theory

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Discussiones Mathematicae - General Algebra and Applications
2013/33/2
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Pages:
167-199
Main language of publication
English
Received
2013-04-09
Published
2013

DOI: 10.7151/dmgaa.1200

Exact and natural sciences