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# Communications in Number Theory and Physics

## Volume 10 (2016)

### Number 1

### A few $c_2$ invariants of circulant graphs

Pages: 63 – 86

DOI: http://dx.doi.org/10.4310/CNTP.2016.v10.n1.a3

#### Author

#### Abstract

The $c_2$ invariant is an arithmetic graph invariant introduced by Schnetz [13] and further developed by Brown and Schnetz [6] 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.