Graph invariants from ideas in physics and number theory
Abstract
We study free scalar field theory on a graph, which gives rise to a modified version of the discrete Green's function on a graph studied in [8]. We show that this gives rise to a graph invariant, which is closely related to the 2-dimensional Weisfeiler-Lehman algorithm for graph isomorphism testing. We then consider the same theory over the integers, which leads to the consideration of certain quadratic forms over the integers as initiated in [14], associated with the graphs. The quadratic form represented by the combinatorial Laplacian respects a well-behaved wedge sum and appears to capture important graph properties regarding graph embeddings into surfaces, namely the graph genus and the dual graphs.
Fields of Study & Subjects
Topological and Geometric Data Analysis
Graph theory and applications
Limits and Structures in Graph Theory
Publication details
- Journal / source
- Journal of Combinatorics
- Publisher
- International Press of Boston
- Volume
- 17
- Issue
- 3
- Pages
- 389–415
- Publication date
- 1 January 2026
- Language
- EN