{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11323",
  "author_total": 4,
  "authors": [
    {
      "name": "Dongxiu Cai"
    },
    {
      "name": "Zhenbo Chen"
    },
    {
      "name": "Jiasheng Zeng"
    },
    {
      "name": "Xiao-Dong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99k4czaztr56t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11323",
  "fulltext_url": null,
  "id": "99k4czaztr56t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11323",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212314450",
  "pdf_url": "https://arxiv.org/pdf/2609.11323",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99k4czaztr56t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polynomial positivity cones for Coxeter roots and walks in trees",
  "title_text": "Polynomial positivity cones for Coxeter roots and walks in trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/99k4czaztr56t",
  "venue": "arXiv",
  "versions": []
}