On Graphs whose Flow Polynomials have Real Roots Only

  • Fengming Dong
Keywords: Graphs, Chromatic polynomials, Flow polynomials, Roots


Let $G=(V,E)$ be a bridgeless graph. In 2011 Kung and Royle showed that the flow polynomial $F(G,\lambda)$ of $G$ has integral roots only if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether every graph whose flow polynomial has real roots only is the dual of some chordal and plane graph. We conclude that the answer for this problem 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 $P(G,\lambda)$ has real roots only 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.
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