The Automorphism Group of the Spectral Incidence Graph over Finite Fields

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 $Aut(\mathbf{SIG}(\mathbb V_0)) \cong \left(\prod_{\mathcal C\in\mathcal T} S_{\mathcal C}\right) \rtimes P\Gamma L_n(q)$. For $n=2$, we obtain $Aut(\mathbf{SIG}(\mathbb V_0)) \cong \left( \prod_{i=1}^{q+1}S_q\times \prod_{i=1}^{\frac{q(q+1)}{2}}S_q \right) \rtimes S_{q+1}.$