A few $c_2$ invariants of circulant graphs

TL;DR

This paper computes the $c_2$ invariant for certain decompleted circulant graphs, providing explicit formulas and recurrence relations that deepen understanding of their arithmetic properties related to Feynman integrals.

Contribution

It introduces methods to compute the $c_2$ invariant for a broad class of decompleted circulant graphs, extending previous work and offering explicit recurrence relations.

Findings

Computed $c_2$ invariant for $C_n(1,3)$ and $C_{2k+2}(1,k)$ at prime 2.

Derived recurrence relations for $c_2$ invariants of multiple circulant graph families.

Extended the understanding of $c_2$ invariants in the context of Feynman graph analysis.

Abstract

The $c_2$ invariant is an arithmetic graph invariant introduced by Schnetz and developed by Brown and Schnetz in order to better understand Feynman integrals. This document looks at the special case where the graph in question is a 4-regular circulant graph with one vertex removed; call such a graph a decompletion of a circulant graph. The $c_2$ invariant for the prime $2$ is computed in the case of the decompletion of circulant graphs $C_n(1,3)$ and $C_{2k+2}(1,k)$. For any prime $p$ and for the previous two families of circulant graphs along with the further families $C_n(1,4)$, $C_n(1,5)$, $C_n(1,6)$, $C_n(2,3)$, $C_n(2,4)$, $C_n(2,5)$, and $C_n(3,4)$, the same technique gives the $c_2$ invariant of the decompletions as the solution to a finite system of recurrence equations.