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A Graph Integral Formulation of the Circuit Partition Polynomial

Published online by Cambridge University Press:  11 October 2011

CRISTOPHER MOORE  and
ALEXANDER RUSSELL
CRISTOPHER MOORE
Affiliation:
Computer Science Department, University of New Mexico, Albuquerque, NM 87131, USA and The Santa Fe Institute, Santa Fe, NM 87501, USA (e-mail: [email protected])
ALEXANDER RUSSELL
Affiliation:
Department of Computer Science & Engineering, University of Connecticut, Storrs, CT 06269, USA (e-mail: [email protected])
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 911 - 920
Copyright
Copyright © Cambridge University Press 2011

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