{
  "abstract": "We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36622",
  "author_total": 2,
  "authors": [
    {
      "name": "Ximin Wang"
    },
    {
      "name": "Zheng Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cgyev08cfnn2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36622",
  "fulltext_url": null,
  "id": "cgyev08cfnn2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36622",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214983334",
  "pdf_url": "https://arxiv.org/pdf/2609.36622",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cgyev08cfnn2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36622",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36622"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The classification of endpoint-transitive graphs with geodesic condition",
  "title_text": "The classification of endpoint-transitive graphs with geodesic condition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/cgyev08cfnn2r",
  "venue": "arXiv",
  "versions": []
}