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

  • Iain Crump

    Simon Fraser University, Burnaby, Canada
  • Matt DeVos

    Simon Fraser University, Burnaby, Canada
  • Karen Yeats

    Simon Fraser University, Burnaby, Canada

Abstract

A 4-point Feynman diagram in scalar theory is represented by a graph which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of 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 .

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 for which is a multiple of (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 is obtained from a -regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.

Cite this article

Iain Crump, Matt DeVos, Karen Yeats, Period preserving properties of an invariant from the permanent of signed incidence matrices. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), no. 4, pp. 429–454

DOI 10.4171/AIHPD/35