{
  "abstract": "Let $\\mathbb{F}_q$ be a finite field with $q$ elements and let $\\mathbb{V}_0=\\mathbb{F}_q^n$ be the $n$-dimensional vector space over $\\mathbb{F}_q$, for an integer $n\\geq2$. We introduce the \\emph{spectral incidence graph} of $\\mathbb{V}_0$, denoted by $\\mathbf{SIG}(\\mathbb{V}_0)$, a bipartite graph whose two vertex classes consist of the one-dimensional subspaces of $M_n(\\mathbb{F}_q)$ generated by matrices having at least one eigenvector in $\\mathbb{V}_0$, and the one-dimensional subspaces of $\\mathbb{V}_0$ (i.e. the projective space $\\mathbb{P}^{n-1} (\\mathbb{F}_q)$), respectively. A matrix vertex $\\langle M\\rangle$ and a point vertex $\\langle v\\rangle$ are adjacent if and only if $v$ is an eigenvector of $M$. We determine several structural parameters of $\\mathbf{SIG}(\\mathbb{V}_0)$, including its twin classes, connectivity, domination number, diameter, vertex degrees, and number of edges. We show that every automorphism of $\\mathbf{SIG}(\\mathbb{V}_0)$ preserves the two parts of the bipartition. We prove that the kernel of the induced action of $Aut(\\mathbf{SIG}(\\mathbb V_0))$ on the projective part $\\mathbb{P}^{n-1} (\\mathbb{F}_q)$ is precisely $\\prod_{\\mathcal C\\in\\mathcal T} S_{\\mathcal C}$, where $\\mathcal T$ denotes the set of twin classes of matrix vertices and $S_{\\mathcal{C}}$ is the symmetric group on $\\mathcal{C}$. We then determine the full automorphism group of $\\mathbf{SIG}(\\mathbb{V}_0)$. For $n\\geq3$, we prove that $Aut(\\mathbf{SIG}(\\mathbb V_0)) \\cong \\left(\\prod_{\\mathcal C\\in\\mathcal T} S_{\\mathcal C}\\right) \\rtimes PΓL_n(q)$. For $n=2$, we obtain $Aut(\\mathbf{SIG}(\\mathbb V_0)) \\cong \\left( \\prod_{i=1}^{q+1}S_q\\times \\prod_{i=1}^{\\frac{q(q+1)}{2}}S_q \\right) \\rtimes S_{q+1}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.26320",
  "author_total": 1,
  "authors": [
    {
      "name": "Ali Majidinya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4n385psprfth.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.26320",
  "fulltext_url": null,
  "id": "d4n385psprfth",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.26320",
  "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.RA"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7171834806",
  "pdf_url": "https://arxiv.org/pdf/2607.26320",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d4n385psprfth.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2607.26320",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.26320"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Automorphism Group of the Spectral Incidence Graph over Finite Fields",
  "title_text": "The Automorphism Group of the Spectral Incidence Graph over Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/d4n385psprfth",
  "venue": "arXiv",
  "versions": []
}