Real-rooted flow polynomials have only integral roots

Abstract

Let $G$ be a connected 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 plane graph. In this article, we further prove that if $F(G,\lambda )$ has real roots only, then $G$ is the dual of a chordal plane graph and each root of $F(G,\lambda )$ is an integer in the set $\{1,2,3\}$.