Graph invariants from ideas in physics and number theory

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.