Some Applications of Wagner's Weighted Subgraph Counting Polynomial

Ferenc Bencs,Péter Csikvári,Guus Regts

doi:10.37236/10185

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 absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

Highlights

The investigation of the location of zeros of different partition functions of graphs and hypergraphs is a topic gaining more and more interest. The reason for this is that these partition functions are related to several topics such as statistical physics, combinatorics and computer science
In computer science it is related to the design of efficient approximation algorithms for computing evaluations of partition functions and graph polynomials
A recent approach by Barvinok [4] combined with results from [24] shows that zero-free regions for graph polynomials imply fast algorithms for approximating evaluations when restricted to bounded degree graphs

Summary

Introduction

The investigation of the location of zeros of different partition functions of graphs and hypergraphs is a topic gaining more and more interest. In this note we give two new zero-free regions, one for the anti-ferromagnetic Ising model on line graphs, and one for the edge cover polynomial. Combining the theorem above with Barvinok’s method [4] and the improvement from [24], we obtain a fully polynomial time approximation scheme for approximating the edge cover polynomial on the the complement of the set {−(1 − α)2 | |α| 1} on bounded degree graphs. To do this one needs to interpolate from ‘infinity’. In the final section we close with an open question and some remarks

Preliminaries

The Cardioid-like region

The antiferromagnetic Ising-model on line graphs

Disk around zero

Further remarks