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 $\mathrm{\mid }W(G)\mathrm{\mid }\le k$, 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$F(G,\lambda)$hasatleast9realrootsin$(1,2)$.