The Automorphism Group of the Spectral Incidence Graph over Finite Fields

@misc{majidinya2026automorphism,
  title = {{The Automorphism Group of the Spectral Incidence Graph over Finite Fields}},
  author = {Ali Majidinya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.26320},
  eprint = {2607.26320},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.26320},
}

TY  - UNPB
TI  - The Automorphism Group of the Spectral Incidence Graph over Finite Fields
AU  - Ali Majidinya
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2607.26320
UR  - https://arxiv.org/abs/2607.26320
AB  - Let 𝔽_q be a finite field with q elements and let 𝕍_0=𝔽_q^n be the n-dimensional vector space over 𝔽_q, for an integer n≥2. We introduce the \emph{spectral incidence graph} of 𝕍_0, denoted by SIG(𝕍_0), a bipartite graph whose two vertex classes consist of the one-dimensional subspaces of M_n(𝔽_q) generated by matrices having at least one eigenvector in 𝕍_0, and the one-dimensional subspaces of 𝕍_0 (i.e. the projective space ℙ^n−1 (𝔽_q)), respectively. A matrix vertex ⟨ M⟩ and a point vertex ⟨ v⟩ are adjacent if and only if v is an eigenvector of M. We determine several structural parameters of SIG(𝕍_0), including its twin classes, connectivity, domination number, diameter, vertex degrees, and number of edges. We show that every automorphism of SIG(𝕍_0) preserves the two parts of the bipartition. We prove that the kernel of the induced action of Aut(SIG(𝕍_0)) on the projective part ℙ^n−1 (𝔽_q) is precisely ∏_𝒞∈𝒯 S_𝒞, where 𝒯 denotes the set of twin classes of matrix vertices and S_𝒞 is the symmetric group on 𝒞. We then determine the full automorphism group of SIG(𝕍_0). For n≥3, we prove that Aut(SIG(𝕍_0)) ≅ (∏_𝒞∈𝒯 S_𝒞) rtimes PΓL_n(q). For n=2, we obtain Aut(SIG(𝕍_0)) ≅ ( ∏_i=1^q+1S_q× ∏_i=1^(q(q+1))/2S_q ) rtimes S_q+1.
ER  -