New graph polynomials from the Bethe approximation of the Ising partition function
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.
Cite
@article{arxiv.0908.3850,
title = {New graph polynomials from the Bethe approximation of the Ising partition function},
author = {Yusuke Watanabe and Kenji Fukumizu},
journal= {arXiv preprint arXiv:0908.3850},
year = {2010}
}
Comments
To appear in Combinatorics, Probability & Computing. Revised from the first version, 28 pages