Period preserving properties of an invariant from the permanent of signed incidence matrices

Abstract

A 4-point Feynman diagram in scalar ${\varphi }^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is invariant under a number of meaningful graph operations – namely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex which was deleted to form $G$.

In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam [1]. The graph permanent applies to any graph $G=(V,E)$ for which $|E|$ is a multiple of $|V|-1$ (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when $G$ is obtained from a $2k$-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.