A Graph Integral Formulation of the Circuit Partition Polynomial

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 a€ˆxu, xva€‰. 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.

[1]  G. C. Wick The Evaluation of the Collision Matrix , 1950 .

[2]  Dirk L. Vertigan,et al.  The Computational Complexity of Tutte Invariants for Planar Graphs , 2005, SIAM J. Comput..

[3]  Hans Wenzl,et al.  On the structure of Brauer’s centralizer algebras , 1988 .

[4]  Joanna A. Ellis-Monaghan New Results for the Martin Polynomial , 1998, J. Comb. Theory, Ser. B.

[5]  François Jaeger,et al.  On Tutte polynomials and cycles of plane graphs , 1987, J. Comb. Theory, Ser. B.

[6]  André Bouchet,et al.  Tutte-martin polynomials and orienting vectors of isotropic systems , 1991, Graphs Comb..

[7]  Béla Bollobás,et al.  The interlace polynomial: a new graph polynomial , 2000, SODA '00.

[8]  Joanna A. Ellis-Monaghan,et al.  Distance Hereditary Graphs and the Interlace Polynomial , 2006, Combinatorics, Probability and Computing.

[9]  Pierre Martin,et al.  Enumérations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck , 1977 .

[10]  L. Isserlis ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES , 1918 .

[11]  Béla Bollobás,et al.  Evaluations of the Circuit Partition Polynomial , 2002, J. Comb. Theory, Ser. B.

[12]  Richard Brauer,et al.  On Algebras Which are Connected with the Semisimple Continuous Groups , 1937 .

[13]  Peter Winkler,et al.  Counting Eulerian Circuits is #P-Complete , 2005, ALENEX/ANALCO.

[14]  Andrea D. Austin,et al.  The Circuit Partition Polynomial with Applications and Relation to the Tutte and Interlace Polynomials , 2007 .

[15]  Michel LasVergnas,et al.  On the evaluation at (3,3) of the Tutte polynomial of a graph , 1988 .