On integer radii coin representations of the wheel graph

TL;DR

This paper explores the geometric and algebraic properties of flower coin graphs representing wheel graphs with integer radii, deriving unique polynomials and parameterizations for their radii.

Contribution

It introduces a unique irreducible polynomial for each n-petaled flower and develops a parameterization for integer radii in the 3-petal case.

Findings

Existence of a unique irreducible polynomial P_n for each n-petaled flower.

A recursion relation satisfied by the polynomials P_n.

A parameterization for all integer radii in the 3-petal flower case.

Abstract

A {\em flower} is a coin graph representation of the wheel graph. A {\em petal} of the wheel graph is an edge to the center vertex. In this paper we investigate flowers whose coins have integer radii. For an $n$-petaled flower we show there is a unique irreducible polynomial $P_n$ in $n$ variables over the integers $\ints$, the affine variety of which contains the cosines of the internal angles formed by the petals of the flower. We also establish a recursion that these irreducible polynomials satisfy. Using the polynomials $P_n$, we develop a parameterization for all the integer radii of the coins of the 3-petal flower.