Bounds for the coefficients of flow polynomials

Abstract

Let 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 $F(G,t)\in \{{(t-1)}^2,(t-1)(t-2)\}$ if $m=n+1$. This paper shows that if $m⩾n+2$, then the absolute value of the coefficient of $t^i$ in the expansion of $F(G,t)$ is bounded above by the coefficient of $t^i$ in the expansion of $(t+1)(t+2)(t+3){(t+4)}^{m-n-2}$ for each i with $0⩽i⩽m-n+1$.