Some applications of Wagner's weighted subgraph counting polynomial

TL;DR

This paper applies Wagner's polynomial to analyze the roots of certain graph polynomials, demonstrating their real-rootedness and bounded root length, and discusses implications for approximation algorithms.

Contribution

It introduces new applications of Wagner's polynomial to prove root properties of the anti-ferromagnetic Ising model and edge cover polynomial, linking them to computational methods.

Findings

Partition function of anti-ferromagnetic Ising model on line graphs is real rooted

Roots of the edge cover polynomial have length at most 4

Results suggest potential for efficient approximation algorithms

Abstract

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have length at most $4$. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.