AbstractLet G be any connected bridgeless (n,m)-graph which may have loops and multiedges. It is known that the flow polynomial F(G,t) of G is a polynomial of degree m−n+1; F(G,t)=t−1 if m=n; and F(G,t)∈{(t−1)2,(t−1)(t−2)} if m=n+1. This paper shows that if m⩾n+2, then the absolute value of the coefficient of ti in the expansion of F(G,t) is bounded above by the coefficient of ti in the expansion of (t+1)(t+2)(t+3)(t+4)m−n−2 for each i with 0⩽i⩽m−n+1.