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On Extremal Coin Graphs, Flowers, and Their Rational Representations

Show simple item record Dunham, Jill Bigley
dc.creator Dunham, Jill Bigley 2009-04-23 2009-07-06T14:43:15Z NO_RESTRICTION en 2009-07-06T14:43:15Z 2009-07-06T14:43:15Z
dc.description.abstract We study extremal coin graphs in the Euclidean plane on n vertices with the maximum number of edges. This is related to the unit coin graph problem first posed by Erdos in 1946, and considers coin graphs that satisfy certain conditions relating to the ratios of the possible radii of the coins in the graph. A motivating problem is a special case of a coin graph with multiple radii. We explore the algebraic equations describing a flower, the coin graph presentation of a wheel graph, and present a class of irreducible symmetric polynomials that describe the relation of the radii of each flower. These polynomials are then used to fully characterize flowers on four coins, also known as Soddy circles, with rational radii. This yields a free parametrization of all flowers on four coins with rational radii. A similar method is used to characterize all flowers on five coins with rational radii and to describe a large class of solutions for flowers on n coins. en
dc.language.iso en_US en
dc.subject coin graph en
dc.subject Galois theory en
dc.subject diophantine equations en
dc.subject plane graphs en
dc.subject symmetric polynomials en
dc.subject discrete geometry en
dc.title On Extremal Coin Graphs, Flowers, and Their Rational Representations en
dc.type Dissertation en Doctor of Philosophy in Mathematics en Doctoral en Mathematics en George Mason University en

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