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# Annales de l’Institut Henri Poincaré D

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**Volume 3, Issue 4, 2016, pp. 429–454**

**DOI: 10.4171/AIHPD/35**

Published online: 2016-12-22

Period preserving properties of an invariant from the permanent of signed incidence matrices

Iain Crump^{[1]}, Matt DeVos

^{[2]}and Karen Yeats

^{[3]}(1) Simon Fraser University, Burnaby, Canada

(2) Simon Fraser University, Burnaby, Canada

(3) Simon Fraser University, Burnaby, Canada

A 4-point Feynman diagram in scalar $\phi^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is invariant under a number of meaningful graph operations – namely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex which was deleted to form $G$.

In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam [1]. The graph permanent applies to any graph $G = (V,E)$ for which $|E|$ is a multiple of $|V| - 1$ (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when $G$ is obtained from a $2k$-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.

*Keywords: *Permanent, Feynman graph, Feynman period

Crump Iain, DeVos Matt, Yeats Karen: Period preserving properties of an invariant from the permanent of signed incidence matrices. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 3 (2016), 429-454. doi: 10.4171/AIHPD/35