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arXiv · 0908.3850

New graph polynomials from the Bethe approximation of the Ising partition function

Yusuke Watanabe·Kenji Fukumizu
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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 (1947, Proc. Cambridge Philos. Soc. 43, 26-40). 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 parameterized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.

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Yusuke Watanabe, Kenji Fukumizu. 2010-06-04. New graph polynomials from the Bethe approximation of the Ising partition function. https://arxiv.org/abs/0908.3850

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