A Graph Integral Formulation of the Circuit Partition Polynomial

Abstract

We present a simple graph integral equivalent to a multiple of the circuit partition polynomial. Let G be a directed graph, and let k be a positive integer. Associate with each vertex v of G an independent, uniformly random k-dimensional complex vector xv of unit length. We define q(G;k) to be the expected value of the product, over all edges (u, v), of the inner product 〈xu, xv〉. We show that q(G;k) is proportional to G's cycle partition polynomial, and therefore that computing q(G;k) is #P-complete for any k > 1. We also study the natural variants that arise when the xv are real or drawn from the Gaussian distribution.