ABSTRACT
Since the focus of this book is on representing graphs as Laplacian matrices, we dedicate this chapter to the foundations of Laplacian matrices. The goal of this chapter is to provide an overview of Laplacian matrices before drilling into more specific and rigorous topics in later chapters. We begin by providing a context of Laplacian matrices in their relationship to other matrix representations of graphs. We then proceed to the Matrix Tree Theorem which is a theorem that first motivated the use of Laplacian matrices in graph theory. Since graphs are discrete objects, we provide further context of Laplacian matrices by investigating the Laplace operator which is the continuous version of the Laplacian matrix. The Laplace operator has applications in energy flow, and we adapt these applications to graphs and learn how to draw graphs which minimize energy. With this in mind, we close this section by applying Laplacian matrices to network flow. Historically, network flow has been a major motivation to studying Laplacian matrices.
