Improved bounds for the coefficient of flow polynomials

TL;DR

This work refines coefficient bounds for flow polynomials $F(G,t)$ of connected bridgeless $(n,m)$-graphs, providing sharp piecewise upper bounds on $|c_i|$ via $d_i$ based on the excess $m-n$, and establishing lower bounds for cubic graphs with only real flow roots using $b_i$ from $(t+1)(t+2)^{n/2}$ (with a refined form for simple graphs). The key techniques combine the $\tau$-transformation, Tutte relations, and a 2-edge-cut decomposition to reduce to cubic or 3-edge-connected components, enabling precise comparisons and tight extremal cases (e.g., graphs like $G^{*}$). The results yield both general bounds and structure-dependent lower bounds, with equality cases fully characterized in the real-root cubic setting and verified on small bridgeless cubic graphs. These findings deepen understanding of flow polynomial coefficients and provide sharp, applicable bounds for extremal graph families in flow theory.

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 $(t+1)(t+2)^{\frac{n}{2}-2}(t+3)^{2}$.

Figures (5)

• Figure 1: Connected bridgeless cubic $(n,m)$-graphs $G_{1}$ and $G_{2}$.
• Figure 2: Graph $G^{*}$.
• Figure 3: 2-edge-cut decomposition operation of $G$
• Figure 4: Partial graph of near-cubic graph at $x$.
• Figure 5: All cubic graphs with $n = 6$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4