$c_2$ invariants of recursive families of graphs

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 .