Spectral graph theory is a branch of mathematics that studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with them, such as the adjacency matrix or the Laplacian matrix. These matrices capture important structural information about the graph, allowing researchers to analyze various characteristics like connectivity, clustering, and the presence of certain subgraphs. By examining the spectrum, or set of eigenvalues, of these matrices, one can gain insights into the graph's behavior and properties. Applications of spectral graph theory include network analysis, chemistry, and computer science, where it helps in understanding complex systems and optimizing algorithms.