Automorphism Group of the Spectral Incidence Graph over Finite Fields

Let $q\geq n\geq 2$ be integers, $\mathbb{F}_q$ a finite field with $q$ elements and $\mathbb{V}_0= \mathbb{F}_q^n$ the $n$-dimensional vector space over $\mathbb{F}_q$. Let $E_n(\mathbb{F}_q)$ denotes the set of all nonzero $n\times n$ matrices over $\mathbb{F}_q$ having an eigenvector. 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 the matrices in $E_n(\mathbb{F}_q)$ and the one-dimensional subspaces of $\mathbb{V}_0$, respectively. A matrix vertex $\langle M\rangle$ is adjacent to a one-dimensional subspace $\langle v\rangle$ of $\mathbb{V}_0$, 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 $\mathbb{V}_0$. Using split short exact sequence theorem for the groups and fundamental theorem of projective geometry we determine the automorphism group of $\mathbf{SIG}(\mathbb V_0)$. For $n\geq3$, we prove that $$\operatorname{Aut}(\mathbf{SIG}(\mathbb V_0)) \cong \left(\prod_{\mathcal C\in\mathcal T} S_{\mathcal C}\right) \rtimes P\Gamma L(n,q),$$ and for $n=2$, we obtain $$\operatorname{Aut}(\mathbf{SIG}(\mathbb V_0)) \cong \left( \prod_{i=1}^{q+1}S_q\times \prod_{i=1}^{\frac{q(q+1)}{2}}S_q \right) \rtimes S_{q+1}.$$ 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 $\mathbf{SIG}(\mathbb V_0)$, including classes of twin points, connectivity, domination number, diameter, vertices degrees and the number of edges.