Some Combinatorial Problems Involving Total Non-Negativity and Unimodality

In this dissertation, we explore two combinatorial problems. The first focuses on totally non-negative matrices (matrices whose minors are all nonnegative). In Chapter 2, we consider a family of change-of-basis matrices, and characterize those that are totally non-negative. This answers a question of Galvin and Pacurar. Then in Chapter 3 we delve into an open problem (due to Brenti) regarding the total non-negativity of the Eulerian matrix and obtain a partial total non-negativity result not only for the Eulerian matrix, but also for a two-variable generalization of the Eulerian matrix introduced recently by Chen, Deb, Dyachenko, Gilmore and Sokal. The second problem addresses a recent conjecture (due to Alikhani and Peng) that the domination sequences of all finite graphs are unimodal. In Chapter 4, we present evidence supporting this conjecture by proving that the domination sequences of arbitrary powers of paths and cycles are indeed unimodal. In addition, in Chapter 5, we highlight some interesting combinatorics of the domination sequences of paths, which leads to other observations and conjectures.