# Graph Invariants with Connections to the Feynman Period in $\phi^4$ Theory

## TL;DR

This paper explores graph invariants related to Feynman periods in $$ theory, introducing the extended graph permanent and analyzing its invariance properties, with implications for understanding the structure of Feynman graphs.

## Contribution

The paper introduces the extended graph permanent, demonstrating its invariance under key graph operations and providing new computational and interpretative methods.

## Key findings

- Extended graph permanent is preserved by key graph operations.
- The sequence of residues from the extended graph permanent has a product property.
- Alternative interpretation as point count of a novel graph polynomial.

## Abstract

Feynman diagrams in $\phi^4$ theory have as their underlying structure 4-regular graphs. In particular, any 4-point $\phi^4$ graph can be uniquely derived from a 4-regular graph by deleting a vertex. The Feynman period is a simplified version of the Feynman integral, and is of special interest, as it maintains much of the important number theoretic information from the Feynman integral. It is also of structural interest, as it is known to be preserved by a number of graph theoretic operations. In particular, the vertex deleted in constructing the 4-point graph does not affect the Feynman period, and it is invariant under planar duality and the Schnetz twist, an operation that redirects edges incident to a 4-vertex cut. Further, a 4-regular graph may be produced by a 3-sum operation on triangles in two 4-regular graphs. The Feynman period of this graph with a vertex deleted is equal to the product of the Feynman periods of the two smaller graphs with one vertex deleted each. These operations currently explain all known instances of non-isomorphic 4-point $\phi^4$ graphs with equal periods. With this in mind, other graph invariants that are preserved by these operations for 4-point $\phi^4$ graphs are of interest, as they may provide insight into the Feynman period. In this thesis the extended graph permanent is introduced; an infinite sequence of residues from prime order finite fields. It is shown that this sequence is preserved by these three operations, and has a product property. Additionally, computational techniques will be established, and an alternate interpretation will be presented as the point count of a novel graph polynomial. Further, the previously existing $c_2$ invariant and Hepp bound are examined, two graph invariants that are conjectured to be preserved by these graph operations. A combinatorial approach to the $c_2$ invariant is introduced.