Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices

TL;DR

This paper investigates the graph permanent, an invariant related to Feynman diagrams in scalar 4 theory, demonstrating its invariance under certain graph operations and its independence from vertex choice in specific cases.

Contribution

It establishes the invariance of the graph permanent under planar duality and the Schnetz twist, and shows its independence from the choice of deleted vertex in certain graphs.

Findings

Graph permanent is invariant under planar duality.

Graph permanent remains unchanged under the Schnetz twist.

Independence from the choice of deleted vertex in 2k-regular graphs.

Abstract

A 4-point Feynman diagram in scalar $\phi^4$ 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. 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.