c_2 Invariants of Recursive Families of Graphs

TL;DR

This paper investigates the c_2 invariant of certain graph families, showing it is zero for decompleted non-skew toroidal grids at p=2 and calculating it for X-ladders, demonstrating applicability to recursive graphs.

Contribution

It provides new results on the c_2 invariant for specific graph families and introduces methods applicable to recursive graphs at fixed primes.

Findings

c_2 invariant is zero for all decompleted non-skew toroidal grids at p=2

Calculated c_2 invariant at p=2 for X-ladders

Methods extend to any recursive graph structure at fixed p

Abstract

The c_2 invariant, defined by Schnetz in 2011, is an arithmetic graph invariant created towards a better understanding of Feynman integrals. This paper looks at some graph families of interest, with a focus on decompleted toroidal grids. Specifically, the c_2 invariant for p=2 is shown to be zero for all decompleted non-skew toroidal grids. We also calculate the c_2 invariant at p=2 for G a family of graphs called X-ladders. Finally, we show these methods can be applied to any graph with a recursive structure, for any fixed p.