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On Zero-free Intervals of Flow Polynomials

This article studies real roots of the flow polynomial $F(G,λ)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $ξ_k$ be the supremum in $(1,2]$ such that $F(G,λ)$ has no real roots in $(1,ξ_k)$ 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 $ξ_k$ can be determined by considering a finite set of graphs and show that $ξ_k=2$ for $k\le 2$, $ξ_3=1.430\cdots$, $ξ_4=1.361\cdots$ and $ξ_5=1.317\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,λ)$ are real but some of these roots are not in the set $\{1,2,3\}$, then $|E|\ge |V|+17$ and $F(G,λ)$ has at least 9 real roots in $(1,2)$.