发表机构
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications; Stockholm University; Guangdong Technion–Israel Institute of Technology; University of Oregon（燕京数学科学应用研究院; 斯德哥尔摩大学; 广东以色列理工学院; 俄勒冈大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究环多项式L_G，证明其包含Tutte多项式并统一多个图不变量，提出与U多项式区分能力等价的猜想，并解决Merino和Noble的开放问题。
AI 中文摘要
我们研究环多项式L_G，这是一个由双区带图代数产生的多元图不变量，通过删除-环-收缩递归定义，其中收缩的边变为环。我们证明L_G包含Tutte多项式，并具有类似的生成森林活动展开。它还确定了无环图的Stanley色对称函数、度序列、诱导边数分布和不独立多项式，以及简单图的团多项式和匹配多项式。将每个森林分量的大小和外部活动分开，得到精细环多项式，我们证明其等价于Noble和Welsh的扩展U多项式以及扩展多色数。这一共同细化的不同特化给出普通U多项式、Tutte的普适V函数和Stanley的Tutte对称函数，将这些不变量置于单一框架中。我们猜想L_G和U多项式在简单图上具有相同的区分能力，并验证了所有至多11个顶点的图。简单性至关重要：我们找到了两个无环多重图，它们具有相同的U多项式但不同的环多项式。它们也具有不同的扩展U多项式，因此普通U多项式不能确定无环多重图上的扩展U多项式。这解决了Merino和Noble的一个开放问题。对于外部双区带代数的得分多胞体P_G，环删除-收缩从格点计数提升到多胞体本身。这给出了森林索引的几何停车复形，其格点划分P_G的格点，并由区间乘积分段线性参数化，区间长度是L_G森林展开中的分量权重。
英文摘要
We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graph algebras and defined by a deletion-loopy-contraction recursion, in which the contracted edge becomes a loop. We show that L_G contains the Tutte polynomial and admits a similar spanning-forest activity expansion. It also determines Stanley's chromatic symmetric function, the degree sequence, the induced edge count profile, and the independence polynomial for loopless graphs, and the clique and matching polynomials for simple graphs. Separating the size and the external activity of each forest component leads to a refined loopy polynomial, which we show to be equivalent to the extended U-polynomial of Noble and Welsh and to the extended polychromate. Different specializations of this common refinement give the ordinary U-polynomial, Tutte's universal V-function, and Stanley's Tutte symmetric function, placing these invariants into a single framework. We conjecture that L_G and the U-polynomial have the same distinguishing power on simple graphs, and verify this for all graphs on at most 11 vertices. Simplicity is essential: we found two loopless multigraphs with equal U-polynomials but distinct loopy polynomials. They also have distinct extended U-polynomials, so the ordinary U-polynomial does not determine the extended one on loopless multigraphs. This solves an open problem by Merino and Noble. For the score polytope P_G of the external bizonotopal algebra, loopy deletion-contraction lifts from the lattice-point enumerator to the polytope itself. This gives forest-indexed geometric parking complexes whose lattice points partition those of P_G, and which are piecewise-linearly parametrized by products of intervals whose lengths are the component weights in the forest expansion of L_G.
Comments67 pages
