On Zero-free Intervals of Flow Polynomials

This article studies real roots of the flow polynomial $F(G,\lambda)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $\xi_k$ be the supremum in $(1,2]$ such that $F(G,\lambda)$ has no real roots in $(1,\xi_k)$ for all graphs $G$ with $|W(G)|\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $\xi_k$ can be determined by considering a finite set of graphs and show that $\xi_k=2$ for $k\le 2$, $\xi_3=1.430\cdots$, $\xi_4=1.361\cdots$ and $\xi_5=1.317\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,\lambda)$ are real but some of these roots are not in the set $\{1,2,3\}$, then $|E|\ge |V|+17$ and $F(G,\lambda)$ has at least 9 real roots in $(1,2)$.