Date of Award
5-1-2026
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Lauderdale, Lindsey-Kay
Abstract
Graphs associated with algebraic structures provide a useful way to study algebraic properties through combinatorial methods. In recent years, several graph constructions arising from groups, such as the power graph and enhanced power graph, have been studied extensively. Motivated by these developments, this dissertation investigates a generalization of these ideas in the setting of finite loops. We begin by introducing the background material needed for the remainder of the work, including definitions and basic properties of loops and quasigroups, along with relevant concepts from graph theory. We also review several graph constructions originally developed for groups and show some results, including a strong structural result, can be generalized beyond groups. The focus of the final chapter is the introduction of a new graph associated with a loop, called the \emph{diassociative graph}. For a finite loop, this graph records which pairs of elements generate associative subloops. We explore basic structural properties of these graphs and examine examples arising from several loop constructions. These results provide a first step toward understanding how graph constructions can be used to study loops that are nearly diassociative. In addition, we introduce a numerical invariant that measures how far a loop is from being diassociative and discuss some initial bounds and examples.
Access
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