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New Graph Polynomials from the Bethe Approximation of the Ising Partition Function

Published online by Cambridge University Press:  30 June 2010

Y. WATANABE  and
K. FUKUMIZU
Y. WATANABE
Affiliation:
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan (e-mail: watay@ism.ac.jp, fukumizu@ism.ac.jp)
K. FUKUMIZU
Affiliation:
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan (e-mail: watay@ism.ac.jp, fukumizu@ism.ac.jp)
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Abstract

We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion–contraction relations and are new examples of the V-function, which was introduced by Tutte (Proc. Cambridge Philos. Soc.43, 1947, p. 26). For these polynomials, we discuss the interpretations of special values and then obtain the bound on the number of sub-coregraphs, i.e., spanning subgraphs with no vertices of degree one. It is proved that the polynomial of one variable is equal to the monomer–dimer partition function with weights parametrized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 299 - 320
Copyright
Copyright © Cambridge University Press 2010

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