On zero-free intervals of flow polynomials

For any integer k≥0, let ξk be the supremum in (1,2] such that the flow polynomial F(G,λ) has no real roots in (1,ξk) for all graphs G with at most k vertices of degrees larger than 3. We prove that ξk can be determined by considering a finite set of graphs and show that ξk=2 for k≤2, ξ3=1.430⋯ , ξ4=1.361⋯ and ξ5=1.317⋯ .