一阶导数色对称重构及其在真树上的应用
First-Derivative Chromatic Symmetric Reconstruction For Proper Trees
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中文总结 AI 辅助
本文研究树的色对称函数能否决定树同构类，通过引入一阶导数不变量并证明其在特定条件下可区分真树，同时给出 Stanley 问题的等价形式及生成森林计数的区分界限。
中文摘要 AI 辅助
设 $T$ 为一棵树。Stanley 提出了色对称函数 $X_T$ 是否在同构意义下决定 $T$ 的问题。我们通过将 $X_T$ 视为幂和对称函数 $p_1, p_2, \dots$ 的多项式，并研究不变量 $\Phi_T = (\partial X_T/\partial p_1)|_{p_1 = 0}$ 来探讨这一开放问题。我们证明，若一棵真树的加权骨架（即通过加权收缩所有叶边得到的树）在非叶顶点处具有不同的权重，则 $\Phi_T$ 能够区分该树。我们进一步证明了 Stanley 问题的一个等价表述，该表述通过在树的每个顶点上附加固定数量的叶子得到。最后，我们计算了至多含 $t$ 条边的生成森林的数量，并按连通分量的大小对其进行分组。我们证明，除非 $t \ge \lfloor k/2 \rfloor$，否则这些计数无法区分 $k \ge 4$ 个顶点上的所有树。
英文摘要
Let $T$ be a tree. Stanley asked whether the chromatic symmetric function $X_T$ determines $T$ up to isomorphism. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \dots$ and studying the invariant $Φ_T = (\partial X_T/\partial p_1)|_{p_1 = 0}$. We prove that $Φ_T$ distinguishes every proper tree whose weighted skeleton, the tree obtained from $T$ by weighted contraction of all leaf edges, has distinct weights at non-leaf vertices. We prove further an equivalent formulation of Stanley's question obtained by attaching a fixed positive number of leaves to every vertex of a tree. Finally, we count spanning forests with at most $t$ edges, grouping them by the sizes of their connected components. We prove that these counts cannot distinguish all trees on $k \ge 4$ vertices unless $t \ge \lfloor k/2 \rfloor$.
发表机构
- University of Kansas（堪萨斯大学）
机构由 AI 辅助整理，请以论文原文为准。
