Spectral graph theory

Spectral graph theory studies how the properties of a graph—comprising nodes (vertices) and connections (edges)—are reflected in the mathematical features of matrices associated with it, like the Laplacian matrix. By examining the eigenvalues and eigenvectors of these matrices, we can uncover insights about the graph's structure, such as connectivity, clusters, and flow patterns. This approach provides a powerful way to analyze complex networks mathematically, with applications ranging from data science and physics to social and biological systems.