This article studies real roots of the flow polynomial F(G,λ) of a
bridgeless graph G. For any integer k≥0, let ξk​ be the supremum in
(1,2] such that F(G,λ) has no real roots in (1,ξk​) for all
graphs G with ∣W(G)∣≤k, where W(G) is the set of vertices in G 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⋯. 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∣≥∣V∣+17 and F(G,λ) has at least 9 real roots in (1,2).Comment: 26 pages, 7 figure