2026/07/28 by Ali Majidinya
Mathematics · #acm:05C25 #acm:15A18 #acm:20B25 #acm:20G40 #acm:51E20 #math.RA #msc:05C25 #msc:15A18 #msc:20B25 #msc:20G40 #msc:51E20
34 pages
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Let q≥ n≥ 2 be integers, \mathbbFq a finite field with q elements and \mathbbV0= \mathbbFqn the n-dimensional vector space over \mathbbFq. Let En(\mathbbFq) denotes the set of all nonzero n× n matrices over \mathbbFq having an eigenvector. We introduce the spectral incidence graph of \mathbbV0 denoted by SIG(\mathbbV0), a bipartite graph whose two vertex classes consist of the one-dimensional subspaces of Mn(\mathbbFq) generated by the matrices in En(\mathbbFq) and the one-dimensional subspaces of \mathbbV0, respectively. A matrix vertex ⟨ M⟩ is adjacent to a one-dimensional subspace ⟨ v⟩ of \mathbbV0, if and only if v is an eigenvector of M. Thus, adjacency is defined by the incidence relation between matrices and their invariant one-dimensional subspaces of \mathbbV0. Using split short exact sequence theorem for the groups and fundamental theorem of projective geometry we determine the automorphism group of SIG(\mathbb V0). For n≥3, we prove that Aut(SIG(\mathbb V0)) ≅ (∏\mathcal C∈\mathcal T S\mathcal C) \rtimes PΓL(n,q), and for n=2, we obtain Aut(SIG(\mathbb V0)) ≅ ( ∏i=1q+1Sq× ∏i=1(q(q+1))/(2)Sq ) \rtimes Sq+1. In both cases, the first factor corresponds to permutations of the classes of twin points (vertices having the same neighborhood). We also determined several structural parameters of SIG(\mathbb V0), including classes of twin points, connectivity, domination number, diameter, vertices degrees and the number of edges.