On graphs whose flow polynomials have real roots only

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.