Improved bounds for the coefficient of flow polynomials

TL;DR

This paper improves bounds on the coefficients of flow polynomials for certain graphs and explores properties of graphs with real flow roots, providing tighter inequalities and specific cases.

Contribution

It refines existing bounds for flow polynomial coefficients and establishes new inequalities for graphs with real flow roots, especially for cubic graphs.

Findings

New upper bounds for coefficients when m ≤ n+3 and m ≥ n+4.

Comparison of coefficients for graphs with real flow roots.

Specific bounds for simple connected bridgeless cubic graphs.

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 $t^{i}$ 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 $t^{i}$ in the expansion $\prod\limits_{j=1}^{m-n+1}(t+j)$; and when $m\geqslant n+4$, $|c_{i}|\leqslant d_{i}$, with $d_{i}$ being the coefficient of $t^{i}$ in the expansion $(t+1)(t+2)(t+3)^{2}(t+4)^{m-n-3}$. 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 $t^{i}$ in the expansion $(t+1)(t+2)^{\frac{n}{2}}$. Notably, if $G$ is simple connected bridgeless cubic graph with only real flow roots, then $b_{i}$ is the coefficient of $t^{i}$ in the expansion $(t+1)(t+2)^{\frac{n}{2}-2}(t+3)^{2}$.