{
  "abstract": "Characterizing graphs uniquely determined by their spectra (DS) is a core open problem in spectral graph theory. While this problem has been extensively investigated for simple undirected graphs, it remains relatively underexplored for oriented graphs. For a simple undirected graph $G$ equipped with an orientation $σ$, the corresponding oriented graph $Σ=(G,σ)$ is the digraph obtained by orienting each edge of $G$ according to $σ$. An oriented graph $Σ$ is said to be \\emph{determined by its generalized skew spectrum} (DGSS) if every oriented graph sharing the same generalized skew spectrum is isomorphic to $Σ$. This paper develops a new sufficient criterion for recognizing DGSS controllable oriented graphs, which applies to a much broader family of graphs than previously known results. Let $S$ be the skew-adjacency matrix of $Σ$, $W(Σ)=[e,Se,\\ldots,S^{n-1}e]$, and $d_n$ the last invariant factor of $W(Σ)$. For each odd prime $p$, we define the polynomial $Φ_p(Σ;x)=\\gcd(χ(S;x),χ(S+J;x))$ over the finite field $\\mathbb{F}_p$, which is invariant under generalized skew cospectrality. By analyzing the square-free part of $Φ_p(Σ;x)$ and the associated $p$-main polynomial, we establish a DGSS sufficient condition under the square-free assumption on $d_n$. The proposed criterion allows higher $p$-nullity and recovers the square-free determinant criterion of Qiu, Wang and Wang~(2019) as a special case. We further provide illustrative examples to verify the wider applicability of our new condition and to highlight the role of the compatibility constraints on the irreducible factors of $Φ_p(Σ;x)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03428",
  "author_total": 4,
  "authors": [
    {
      "name": "Limeng Lin"
    },
    {
      "name": "Wei Wang"
    },
    {
      "name": "Wei Wang"
    },
    {
      "name": "Hao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7za8zfxe1pfk9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03428",
  "fulltext_url": null,
  "id": "7za8zfxe1pfk9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03428",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7208793446",
  "pdf_url": "https://arxiv.org/pdf/2609.03428",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7za8zfxe1pfk9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03428",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03428"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A New Sufficient Condition for Oriented Graphs Determined by Their Generalized Skew Spectra",
  "title_text": "A New Sufficient Condition for Oriented Graphs Determined by Their Generalized Skew Spectra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/7za8zfxe1pfk9",
  "venue": "arXiv",
  "versions": []
}