关于串并联偏序集的Tutte多项式

中文摘要 AI 辅助

Tutte多项式是一个双变量多项式，在图论和拟阵论中已被广泛研究。Gordon是第一个研究由偏序集P诱导的贪心拟阵的Tutte多项式T(P;x,y)的人，其中包括串并联偏序集这一特例。特别地，Gordon和McMahon猜想：对于任意两个串并联偏序集P和Q，等式T(P;x,y)=T(Q;x,y)成立当且仅当P同构于Q。在研究这一猜想的过程中，Gordon引入了串并联偏序集的一个子类P，并证明了该等价关系对所有P,Q∈P成立。在本文中，我们引入了串并联偏序集的一个新子类H，并证明了P⊆H。此外，我们证明：对于每一个P∈H和每一个Q∈SP，等式T(P;x,y)=T(Q;x,y)成立当且仅当P同构于Q，从而推广了Gordon的结果。

英文摘要

The Tutte polynomial is a bivariate polynomial that has been extensively studied in graph and matroid theory. Gordon was the first to study the Tutte polynomial $T(P;x,y)$ of the greedoid induced by a poset $P$, including the special case of series-parallel posets. In particular, Gordon and McMahon conjectured that, for any two series-parallel posets $P$ and $Q$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$. In studying this conjecture, Gordon introduced a subclass $\mathcal{P}$ of series-parallel posets and proved that this equivalence holds for all $P,Q\in\mathcal{P}$. In this paper, we introduce a new subclass $\mathrm H$ of series-parallel posets and prove that $\mathcal P\subseteq\mathrm H$. Moreover, we show that, for every $P\in\mathrm H$ and every $Q\in\mathrm{SP}$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$, thereby extending Gordon's result.

发表机构

• School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
• School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）
• Qinghai Institute of Applied Mathematics（青海省应用数学研究所）
• School of Mathematics and System Sciences, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）

机构由 AI 辅助整理，请以论文原文为准。