Date of Submission

Spring 2015

Academic Programs and Concentrations

Mathematics

Project Advisor 1

Maria Belk

Abstract/Artist's Statement

The realizability of a graph is the smallest dimension, d, in which for any realization (placement of the vertices) of the graph in any N-dimensional Euclidean space, there exists a realization of the graph with the same edge lengths in d-dimensional Euclidean space. Expanding on the work of Belk and Connelly who determined the set of all forbidden minors for dimensions up to 3, we determine a large family of forbidden minors for each dimension greater than 3. At the heart of this graph family is a new concept, spherical realizability, which places the vertices of a graph on a d-sphere, rather than in Euclidean space. In addition, we prove theorems regarding rigidity and realizability, and we bound above and below the realizability of certain graph families including bipartite graphs, powers of cycles, and complements of powers of cycles.

Open Access Agreement

On-Campus only

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

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