{
  "abstract": "We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini--Häggström--Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl--Ne\\v set\\v ril--Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19728",
  "author_total": 1,
  "authors": [
    {
      "name": "Yinfeng Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ct4qjxv7pt61.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19728",
  "fulltext_url": null,
  "id": "8ct4qjxv7pt61",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19728",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213582874",
  "pdf_url": "https://arxiv.org/pdf/2609.19728",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ct4qjxv7pt61.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19728",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19728"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Paths maximize the expected range of graph-indexed random walks",
  "title_text": "Paths maximize the expected range of graph-indexed random walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/8ct4qjxv7pt61",
  "venue": "arXiv",
  "versions": []
}