Enumeration of Independent Sets in Graphs

Abstract

An independent set is one of the most natural structures in a graph to focus on, from both a pure and applied perspective. In the realm of graph theory, and any concept it can represent, an independent set is the mathematical way of capturing a set of objects, none of which are related to each other. As graph theory grows, many questions about independent sets are being asked and answered, many of which are concerned with the enumeration of independent sets in graphs. We provide a detailed introduction to general graph theory for those who are not familiar with the subject, and then develop the basic language and notation of independent set theory before cataloging some of the history and major results of the field. We focus particularly on the enumeration of independent sets in various classes of graphs, with the heaviest focus on those defined by maximum and minimum degree restrictions. We provide a brief, specific history of this topic, and present some original results in this area. We then speak about some questions which remain open, and end the work with a conjecture for which we provide strong, original evidence. In the appendices, we cover all other necessary prerequisites for those without a mathematical background