Improved bounds for the coefficient of flow polynomials

Abstract: Let $G$ be a connected bridgeless $(n,m)$-graph which may have loops and multiedges, and let $F(G,t)$ denote the flow polynomial of $G$. Dong and Koh \cite{Dong1} established an upper bound for the absolute value of coefficient $c_{i}$ of $t<sup>{i}$ in the expansion of $F(G,t)$, where $0\leqslant i \leqslant m-n+1$. In this paper, we refine the aforementioned bound. Specifically, we demonstrate that when $n \leqslant m \leqslant n+3$, $|c_{i}|\leqslant d_{i}$, where $d_{i}$ is the coefficient of $t<sup>{i}$ in the expansion $\prod\limits_{j=1}<sup>{m-n+1}(t+j)$; and when $m\geqslant n+4$, $|c_{i}|\leqslant d_{i}$, with $d_{i}$ being the coefficient of $t<sup>{i}$ in the expansion $(t+1)(t+2)(t+3)<sup>{2}(t+4)<sup>{m-n-3}$. Furthermore, we prove that if $G$ is a connected bridgeless cubic graph having only real flow roots, then $b_{i}\leqslant |c_{i}|$, where $b_{i}$ is the coefficient of $t<sup>{i}$ in the expansion $(t+1)(t+2)<sup>{\frac{n}{2}}$. Notably, if $G$ is simple connected bridgeless cubic graph with only real flow roots, then $b_{i}$ is the coefficient of $t<sup>{i}$ in the expansion $(t+1)(t+2)<sup>{\frac{n}{2}-2}(t+3)<sup>{2}$.