Period preserving properties of an invariant from the permanent of signed incidence matrices
Annales de l’Institut Henri Poincaré D, Tome 3 (2016) no. 4, pp. 429-454.

A 4-point Feynman diagram in scalar ϕ 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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/35
Classification : 05-XX, 81-XX
Mots-clés : Permanent, Feynman graph, Feynman period 
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     author = {Crump, Iain and DeVos, Matt and Yeats, Karen},
     title = {Period preserving properties of an invariant from the permanent of signed incidence matrices},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {429--454},
     volume = {3},
     number = {4},
     year = {2016},
     doi = {10.4171/aihpd/35},
     zbl = {1355.05117},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/35/}
}
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Crump, Iain; DeVos, Matt; Yeats, Karen. Period preserving properties of an invariant from the permanent of signed incidence matrices. Annales de l’Institut Henri Poincaré D, Tome 3 (2016) no. 4, pp. 429-454. doi : 10.4171/aihpd/35. http://archive.numdam.org/articles/10.4171/aihpd/35/

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