Date of Award

2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Nancy Eaton

Abstract

Representations of graphs are a way of encoding the structure of a graph by using other discrete structures. The object of a representation of a graph is to encode its structure efficiently. Typically a graph can be encoded by an n x n adjacency matrix. It is possible, however, to encode graphs much more efficiently using other representation schemes.

This work considers tree representations of graphs. A tree representation of a target graph G is an assignment of subtrees of a host tree to the vertices of G in such a way that if uv ∈ E(G), then the subtree assigned to the vertex u and the subtree assigned to the vertex v have at least t nodes in common. This study considers tree representations such that the host tree comes from the family of subdivided n-stars. The largest such representable asteroidal set is constructed, and a lower bound on the length of the longest cycle representable on this family of host tree is also discovered.

Next we move to a different area of study. The study of closure systems and closure spaces is a relatively new direction in mathematics. Jamison writes in his new text that `the notion of closure is pervasive throughout mathematics'. Surely, closure and closed sets can be discussed in almost any mathematical setting. It has been shown by Pfaltz in 1995 that for any finite ground set S, with |S| = n such that n ≥ 10, there are nn unique closure operators on S.

In topology, the separation properties provide criteria for categorizing topological spaces. While not all closure spaces are topological spaces, we may still explore whether the separation properties hold under certain conditions. This work defines a class of closure operators on the integers and investigates the conditions for which the resulting closure space satisfies the different definitions of separability.

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