The Geometric Origins of Spectral Graph Theory

Spectral graph theory is the study of the intimate relationship of eigenvalues to different properties of graphs. Most typically the eigenvalues are associated to eigenfunctions of matrices associated to graphs known as Laplacians. Let $latex G = (V, E)$ be a weighted or unweighted undirected graph (there are simple extensions to directed graphs for many problems). Let $latex A$ be the adjacency matrix and let $latex D$ be the diagonal degree matrix, that is, $latex (D)_{ii} = d_i$, the degree of vertex $latex i$. Then a common definition for the Laplacian is $latex L = D-A$.

As I develop in my research in spectral graph theory, I am consistently amazed by the truth that many results in spectral graph theory can be seen as discrete analogues to results in spectral geometry. I am not accustomed to thinking of graphs as geometric objects, but in fact graph Laplacian matrices are nicely related to Laplace operators on Riemannian manifolds. In this post, I’d like to discuss a few of these relationships. Continue reading