Re-imagining Spectral Graph Theory
Sinan G. Aksoy, Stephen J. Young
TL;DR
The paper introduces the inner product Laplacian, a unified spectral framework defined on general inner product spaces that encapsulates the combinatorial, normalized, hypergraph, and directed Laplacians. It develops a conformality-based theory to relate chosen inner products to natural bases, and uses this to prove generalized isoperimetric results such as a Cheeger-type inequality and an Expander Mixing Lemma for IP Laplacians. It further shows how many higher-order and hypergraph Laplacians arise as IP Laplacians, and demonstrates that Neumann subgraph eigenvalues can be obtained as limits of IP-Laplacian spectra while Dirichlet spectra require more intricate domain restrictions. The framework enables domain-informed fusion of combinatorial structure with nonlocal similarity data, offering a flexible spectral toolkit with potential practical impact in clustering and network analysis, and it highlights computational aspects such as the NP-completeness of computing conformality.
Abstract
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.