Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices

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A 4-point Feynman diagram in scalar ϕ⁴ 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. 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.