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Contributions to Geometry and Graph Theory

In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, the three pairs of continuations of opposite sides meet on a straight line." Our generalization of this theorem will consider a pair of n-simplices intersecting an (n-1)-sphere, and will conclude with the intersections of corresponding faces lying in a hyperplane. In graph theory we will explore the interaction between zero forcing and cut-sets. The color change rule which lies at the center of zero forcing says "Suppose that each of the vertices of a graph are colored either blue or white. If u is a blue vertex and v is its only white neighbor, then u can force v to change to blue." The concept of zero forcing was introduced by the AIM Minimum Rank - Special Graphs Work Group in 2007 as a way of determining bounds on the minimum rank of graphs. Later, Darren Row established results concerning the zero forcing numbers of graphs with a cut-vertex. We will extend his work by considering graphs with arbitrarily large cut-sets, and the collections of components they yield, to determine results for the zero forcing numbers of these graphs.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc1707341
Date08 1900
CreatorsSchuerger, Houston S
ContributorsAnghel, Nicolae, Brand, Neal, Fishman, Lior
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formativ, 73 pages, Text
RightsPublic, Schuerger, Houston S, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved.

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