On graphs whose flow polynomials have real roots only

TL;DR

This paper investigates the conditions under which the flow polynomial of a bridgeless graph has only real roots, revealing a connection to the graph's dual being chordal and plane, and identifying specific root interval properties.

Contribution

It establishes a criterion linking real roots of flow polynomials to the graph's dual being chordal and plane, and characterizes graphs with all real roots based on their structure and roots in (1,2).

Findings

Flow polynomials have only real roots if and only if no roots lie in (1,2) for certain graphs.

Graphs with all real roots are either specific small graphs or have at least 9 roots in (1,2).

The dual of a graph with real roots in its flow polynomial must be chordal and plane under certain conditions.

Abstract

Let $G$ be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial $F(G,\lambda)$ of $G$ are integers if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph $G$ for which $F(G,\lambda)$ has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for $G$ is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then all roots of $P(G,\lambda)$ are real if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.