“Now hold on one minute” you might say. “You just spent all that time talking about graph theory, but why does your project title talk about spectral theory? You must clearly not know what you’re talking about.” To which I would respond that you are absolutely correct, and further would likely ask you why you’re reading a random math blog when you could be enjoying so much more entertaining content, enjoying time outside, or having a delicious meal, but we all have our questions.
Regardless, what is spectral theory, and how does it relate to graph theory? In short, graph theory is the structure of networks and their properties. When representing a graph mathematically, we can do it through the use of its adjacency matrix in matrix form. When in this form, we suddenly have access to countless tools to analyze, transform, and use these matrices to identify properties about the graph and its underlying structure. This is where things get really interesting.
When represented as a matrix, we can then identify the spectral properties of the matrix: its eigenvalues and eigenvectors and what their behavior tells us about the system. This is where we can really start to analyze the graph, and the spectral properties of the eigenvalues and eigenvectors can tell us a lot about the system.