关于简单有根图的Greedoid Tutte多项式
On the Greedoid Tutte Polynomial for simple rooted graphs
AI总结：
本文针对Gordon和McMahon提出的关于简单有根图greedoid Tutte多项式x-adic赋值等于根邻接叶点数的猜想，给出了完整证明，推进了图论与greedoid理论中多项式不变量的研究。
AI中文摘要：
Tutte多项式通过其秩生成公式，为描述图与拟阵的各类组合结构提供了统一框架，是连接图论、拟阵论及其相关应用的重要多项式不变量。设$G=(V(G),E(G),r(G))$为简单有根图，其中$V(G)$是顶点集，$E(G)$是边集，$r(G)$是根。设$Γ(G)$是由有根图$G$诱导的greedoid（广义拟阵），其greedoid Tutte多项式记为$T(Γ(G);x,y)$。令$r=r(G)$，$L_r(G)=\{v\in V(G)\setminus\{r\}: rv\in E(G),\\ d_G(v)=1\}$，且$\ell_r(G)=|L_r(G)|$。Gordon和McMahon提出如下猜想：对于有根图，$\operatorname{ord}_{x}T(Γ(G);x,y)=\ell_r(G)$，其中$\operatorname{ord}_{x}T(Γ(G);x,y)=\max\{a\in\mathbb Z_{\geq0}:x^a\mid T(Γ(G);x,y)\}$，即整除$T(Γ(G);x,y)$的$x$的最高次幂等于与根相邻的叶顶点数目。本文证明了该猜想对所有简单有根图均成立。
英文摘要：
The Tutte polynomial, through its rank-generating formula, provides a unified framework for describing various combinatorial structures of graphs and matroids, and serves as an important polynomial invariant connecting graph theory, matroid theory, and their related applications. Let $G=(V(G),E(G),r(G))$ be a simple rooted graph, where $V(G)$ is the vertex set, $E(G)$ is the edge set, and $r(G)$ is the root. Let $Γ(G)$ be the greedoid induced by the rooted graph $G$, and let its greedoid Tutte polynomial be denoted by $T(Γ(G);x,y)$. Let $r=r(G)$, $L_r(G)=\{v\in V(G)\setminus\{r\}: rv\in E(G),\ d_G(v)=1\}$, and $\ell_r(G)=|L_r(G)|$. Gordon and McMahon proposed the following conjecture: for rooted graphs, $\operatorname{ord}_{x}T(Γ(G);x,y)=\ell_r(G)$, where $\operatorname{ord}_{x}T(Γ(G);x,y)=\max\{a\in\mathbb Z_{\geq0}:x^a\mid T(Γ(G);x,y)\}$, that is, the highest power of $x$ dividing $T(Γ(G);x,y)$ is equal to the number of leaf vertices adjacent to the root. In this paper, we proved that this conjecture holds for all simple rooted graphs.
