{
  "abstract": "For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06747",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhipeng Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/48rr98et76t4d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06747",
  "fulltext_url": null,
  "id": "48rr98et76t4d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06747",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7212081593",
  "pdf_url": "https://arxiv.org/pdf/2609.06747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/48rr98et76t4d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Punctured adjacency-degree algebras of Cartesian products",
  "title_text": "Punctured adjacency-degree algebras of Cartesian products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/48rr98et76t4d",
  "venue": "arXiv",
  "versions": []
}