## All Dissertations

August 2021

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

#### Committee Member

Keri Sather-Wagstaff

James Coykendall

#### Committee Member

Felice Manganiello

Wayne Goddard

#### Abstract

One of the fundamental connections between commutative algebra and graph theory is the relationship between edge ideals and minimal vertex covers of a graph. Let $G=(V,E)$ be a simple graph with $V=\{x_1,\ldots,x_d\}$. The edge ideal of a graph is the ideal $I_G=(x_ix_j) \subsetneq k[x_1,\ldots,x_d]$ generated by the edges $x_ix_j \in E$. A vertex cover graph $G$ is a set of vertices $V' \subset V$ such that every edge in $G$ is incident to a member of $V'$. In 1990, Villarreal showed the edge ideal of a tree Cohen-Macaulay if and only if the tree is unmixed with respect to vertex covers, i.e. all of its minimal vertex covers have the same size.

In this dissertation, we consider two related graph domination problems and study their associated ideals in commutative algebra. The first problem is the PMU placement problem which has its roots in electrical engineering. We prove an analogous result in this setting, namely that the power edge ideal of a tree is Cohen-Macaulay if and only if the tree is unmixed with respect to PMU covers. The second problem is total domination in graph theory for which we define the open neighborhood ideal of a graph. We prove that the open neighborhood ideal of a tree is Cohen-Macaulay if and only if the tree is unmixed with respect to total domination. In each setting, we give a precise characterization of the unmixed trees.

COinS