Coxeter 根的多元正性锥与树中的游走
Polynomial positivity cones for Coxeter roots and walks in trees
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中文总结 AI 辅助
本文证明有限树中游走数量的一个猜想不等式，通过 Coxeter 根的正性锥、生成函数递推和谱协方差恒等式确定等号条件，并推广到带根游走情形。
中文摘要 AI 辅助
对于有限简单图 $G$ 和整数 $k\ge0$，令 $w_k(G)$ 表示长度为 $k$ 的游走数量。我们证明了 Täubig、Weihmann、Kosub、Hemmecke 和 Mayr 对每个有限树的猜想，并确定了所有等号成立的情形。若 $T$ 有 $n\ge1$ 个顶点，则对每个 $k\ge1$ 有 $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$；对于 $n\ge3$，等号成立当且仅当 $T$ 是星图且 $k$ 为偶数，而对于 $n=1$ 或 $n=2$，等号对每个 $k\ge1$ 都成立。对于非 Dynkin 树和偶数指标，证明基于与有限图的邻接算子及其单圈 Coxeter 系统的正实根相关联的多元正性锥。对于有限连通二部非 Dynkin 图，我们建立了关于 Coxeter 轨道和反演集的充分正性条件，并验证了这些条件对支撑在连通导出子树上的指示根成立。对于非 Dynkin 树，这产生了带根的偶数指标不等式，并在求和后得到相应的全局不等式。我们还证明，若 $G$ 是有限连通二部非 Dynkin 简单图，$\varnothing\ne U\subseteq V(G)$，且 $U$ 在 $G$ 中导出的子图是树，则对每个 $k\ge0$ 有 $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$，其中 $w_k(G,U)$ 统计 $G$ 中长度为 $k$ 且起点和终点都在 $U$ 中的游走数量；中间顶点不受限制。有限 Dynkin 树的剩余偶数指标情形通过生成函数递推处理，而奇数指标情形则遵循谱协方差恒等式。
英文摘要
For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of Täubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1$; for $n\ge3$, equality holds if and only if $T$ is a star and $k$ is even, whereas for $n=1$ or $n=2$, equality holds for every $k\ge1$. For non-Dynkin trees and even indices, the proof is based on a polynomial positivity cone associated with the adjacency operator of a finite graph and a positive real root of its simply-laced Coxeter system. For finite connected bipartite non-Dynkin graphs, we establish sufficient positivity conditions in terms of Coxeter orbits and inversion sets, and verify these conditions for indicator roots supported on connected induced subtrees. For non-Dynkin trees, this yields the rooted even-index inequality and, after summation, the corresponding global inequality. We also prove that if $G$ is a finite connected bipartite non-Dynkin simple graph, $\varnothing\ne U\subseteq V(G)$, and the subgraph of $G$ induced by $U$ is a tree, then $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$ for every $k\ge0$, where $w_k(G,U)$ counts the length-$k$ walks in $G$ whose initial and terminal vertices lie in $U$; the intermediate vertices are unrestricted. The remaining even-index cases for finite Dynkin trees are handled by generating-function recurrences, while the odd-index cases follow from a spectral covariance identity.
发表机构
- Shanghai Jiao Tong University（上海交通大学）
- Hong Kong University of Science and Technology（香港科技大学）
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