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Annales de l’Institut Henri Poincaré D

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Volume 6, Issue 2, 2019, pp. 289–311
DOI: 10.4171/AIHPD/72

Published online: 2019-03-20

$c_2$ invariants of recursive families of graphs

Wesley Chorney[1] and Karen Yeats[2]

(1) Simon Fraser University, Burnaby, Canada
(2) University of Waterloo, Canada

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 $p$.

Keywords: $c_2$ invariants, recursive families of graphs, Kirchhoff polynomials, toroidal grids, spanning forest polynomials

Chorney Wesley, Yeats Karen: $c_2$ invariants of recursive families of graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), 289-311. doi: 10.4171/AIHPD/72