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

TL;DR

This paper introduces two new graph polynomials inspired by the Bethe approximation of the Ising model, exploring their properties, relations to known functions, and implications for graph substructures and partition functions.

Contribution

It presents novel graph polynomials satisfying deletion-contraction relations, linking them to the V-function and the monomer-dimer partition function, expanding the theoretical framework.

Findings

Polynomials satisfy deletion-contraction relations.

One polynomial equals the monomer-dimer partition function.

Bound established on the number of sub-coregraphs.

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.