\(c_2\) invariants of recursive families of graphs (Q2421734)

Summary: The \(c_2\) invariant, defined by \textit{O. Schnetz} [Electron. J. Comb. 18, No. 1, Research Paper P102, 23 p. (2011; Zbl 1217.05110)], 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 \(p\).