Physical Sciences and Mathematics Commons^™

Results On Select Combinatorial Problems With An Extremal Nature, Stephen Smith

Theses and Dissertations

This dissertation is split into three sections, each containing new results on a particular combinatorial problem. In the first section, we consider the set of 3-connected quadrangulations on n vertices and the set of 5-connected triangulations on n vertices. In each case, we find the minimum Wiener index of any graph in the given class, and identify graphs that obtain this minimum value. Moreover, we prove that these graphs are unique up to isomorphism.

In the second section, we work with structures emerging from the biological sciences called tanglegrams. In particular, our work pertains to an invariant of tanglegrams called …

A Partial Order On The Orthogonal Group, Colum Watt, Thomas Brady

Articles

We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace.