Re-imagining Spectral Graph Theory

We propose a Laplacian based on general inner product spaces, which we call the inner product Laplacian. We show the combinatorial and normalized graph Laplacians, as well as other Laplacians for hypergraphs and directed graphs, are special cases of the inner product Laplacian. After developing the necessary basic theory for the inner product Laplacian, we establish generalized analogs of key isoperimetric inequalities, including the Cheeger inequality and expander mixing lemma. Dirichlet and Neumann subgraph eigenvalues may also be recovered as appropriate limit points of a sequence of inner product Laplacians. In addition to suggesting a new context through which to examine existing Laplacians, this generalized framework is also flexible in applications: through choice of an inner product on the vertices and edges of a graph, the inner product Laplacian naturally encodes both combinatorial structure and domain-knowledge.