A survey on the study of real zeros of flow polynomials

TL;DR

This survey reviews research on the real zeros of flow polynomials in graph theory, highlighting key results and open problems since their introduction by Tutte in 1950.

Contribution

It compiles and discusses existing results and challenges related to the real zeros of flow polynomials, providing a comprehensive overview of the field.

Findings

Summary of key results on real zeros of flow polynomials

Identification of open problems and research directions

Historical overview of flow polynomial zero studies

Abstract

For a bridgeless graph $G$, its flow polynomial is defined to be the function $F(G,q)$ which counts the number of nonwhere-zero $\Gamma$-flows on an orientation of $G$ whenever $q$ is a positive integer and $\Gamma$ is an additive Abelian group of order $q$. It was introduced by Tutte in 1950 and the locations of zeros of this polynomial have been studied by many researchers. This article gives a survey on the results and problems on the study of real zeros of flow polynomials.