Graduation Year

2016

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Dagan Karp

Reader 2

Charles Doran

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© 2016 Madeleine A Weinstein

Abstract

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding volumes and linear ranks.

Second, we consider the case of a reduced arithmetical graph structure on the hypercube and explore the wealth of relationships that exist between its linear rank and several notions of genus that appear in the literature on graph theory and adinkras.

Third, we study modifications of the definition of an arithmetical graph that incorporate some of the properties of an adinkra, such as the vertex height assignment or the edge dashing. To this end, we introduce the directed arithmetical graph and the dashed arithmetical graph. We then explore properties of these modifications in an attempt to see if our definitions make sense, answering questions such as whether the volume is still an integer and whether there are still only finitely many arithmetical structures on a given graph.

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