Communications in Number Theory and Physics

Volume 11 (2017)

Number 4

Properties of the extended graph permanent

Pages: 791 – 836



Iain Crump (Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada)


We create for all graphs a new invariant, an infinite sequence of residues from prime order finite fields, constructed from the permanent of a reduced incidence matrix. Motivated by a desire to better understand the Feynman period in $\phi^4$ theory, we show that this invariant is preserved by all graph operations known to preserve the period. We further establish properties of this sequence, including computation techniques and alternate interpretations as the point count of a novel polynomial.


extended graph permanent, permanent, Feynman period

2010 Mathematics Subject Classification

Primary 05C50. Secondary 81Q30.

Received 3 January 2017

Accepted 2 June 2017

Published 29 November 2017