$c_2$ invariants of recursive families of graphs

Abstract

The $c_2$ invariant, defined by Schnetz in [17], 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 $c_2^{(2)}(G)$ 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 $p$.