Graph invariants from ideas in physics and number theory

TL;DR

This paper introduces novel graph invariants inspired by physics and number theory, combining Green's functions and quadratic forms to improve graph isomorphism testing methods.

Contribution

It presents a new graph invariant derived from scalar field theory and quadratic forms, offering a fresh approach to the graph isomorphism problem.

Findings

The invariant relates to the 2D Weisfeiler-Lehman algorithm.

Combining invariants enhances graph isomorphism testing.

Provides a bridge between physics, number theory, and graph theory.

Abstract

We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic forms over the integers. We explain that the combination of these two ideas give rise to an interesting approach to the graph isomorphism problem.