{
  "abstract": "A unitary signing of a d-regular graph is a Hermitian adjacency matrix M whose nonzero entries lie in {±1, ±i} and satisfies M 2 = dI. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the Θ-property for a generating set S ⊆ Z n 2 : whenever three pairwise disjoint subsets of S have the same sum, at least two of them have even size.\n\nWe prove that every zero-free generating set with the Θ-property gives a cube-like graph Q S admitting a unitary signing, and we give an explicit local formula for such a signing. For Sidon sets, the Θ-property is also necessary, yielding a characterization of the Sidon cube-like graphs that admit unitary signings. In this setting there are exactly 2 |S|-n switching-equivalence classes of unitary signings, and we characterize when a unitary signing can be chosen to be orthogonal.\n\nThe Θ-property admits a linear-algebraic description in terms of the dependency space D of S:\n\nEquivalently, the map\n\nis linear on D. For the corresponding extremal set problem, in which zero is permitted, this formulation yields the sharp bound |S| ≤ 2n + 1 for generating sets with the Θproperty. We give elementary constructions attaining this bound for every n ≥ 3. Under the additional Sidon condition, the same extremal value is attained for every n ≥ 10 using binary self-dual codes. The exact Sidon maximum is also determined for 3 ≤ n ≤ 9.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33819",
  "author_total": 2,
  "authors": [
    {
      "name": "Meirun Chen"
    },
    {
      "name": "Reza Naserasr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enqcpwbnx2chj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33819",
  "fulltext_url": null,
  "id": "enqcpwbnx2chj",
  "issns": [],
  "landing_url": "https://hal.science/hal-05766226v1",
  "language": "en",
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "hal",
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    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7214745832",
  "pdf_url": "https://hal.science/hal-05766226/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enqcpwbnx2chj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "hal-05766226",
      "source": "hal",
      "url": "https://hal.science/hal-05766226v1"
    },
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Orthogonal and unitary signings of cube-like graphs",
  "title_text": "Orthogonal and unitary signings of cube-like graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/enqcpwbnx2chj",
  "venue": "HAL",
  "versions": []
}