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.

]]>This thesis builds on that project, by improving our ability to identify gene families --- groups of genes in different strains that are related. Previously, similarity was measured only by comparing two genes' DNA sequences, ignoring their positions on the organism's DNA. Here, we leverage genes' relative position to make a better measurement of gene similarity. These improved similarity measurements will improve the existing pipeline's ability to identify HGT events.

]]>Our other results are combinatorial proofs that the conjecture holds for graphs having |*X*| ≤ 2, for even cycles, and under the operation of connecting two graphs by a new edge.

We also make two new conjectures based on empirical data, each of which is stronger than Ehrenborg's conjecture.

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