发表机构
Salman Farsi University of Kazerun（萨尔曼法尔西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了有限域上谱关联图$\mathbf{SIG}(\mathbb{V}_0)$的自同构群，给出$n\geq3$与$n=2$时的不同群结构，同时明确了该图的孪生点类、连通性等结构参数。
AI 中文摘要
设$q\geq n\geq2$为整数，$\mathbb{F}_q$是含$q$个元素的有限域，$\mathbb{V}_0=\mathbb{F}_q^n$是$\mathbb{F}_q$上的$n$维向量空间。记$E_n(\mathbb{F}_q)$为$\mathbb{F}_q$上所有具有特征向量的非零$n\times n$矩阵构成的集合。我们引入$\mathbb{V}_0$的**谱关联图**，记为$\mathbf{SIG}(\mathbb{V}_0)$，它是一个二分图，两个顶点类分别由$E_n(\mathbb{F}_q)$中矩阵生成的$M_n(\mathbb{F}_q)$的一维子空间，以及$\mathbb{V}_0$的一维子空间构成。矩阵顶点$\langle M\rangle$与$\mathbb{V}_0$的一维子空间$\langle v\rangle$相邻当且仅当$v$是$M$的特征向量，即邻接关系由矩阵与其在$\mathbb{V}_0$中的不变一维子空间的关联关系定义。利用群的分裂短正合序列定理与射影几何基本定理，我们确定了$\mathbf{SIG}(\mathbb{V}_0)$的自同构群：当$n\geq3$时，$\operatorname{Aut}(\mathbf{SIG}(\mathbb{V}_0)) \cong \left(\prod_{\mathcal C\in\mathcal T} S_{\mathcal C}\right) \rtimes P\Gamma L(n,q)$；当$n=2$时，$\operatorname{Aut}(\mathbf{SIG}(\mathbb{V}_0)) \cong \left( \prod_{i=1}^{q+1}S_q\times \prod_{i=1}^{\frac{q(q+1)}{2}}S_q \right) \rtimes S_{q+1}$。两种情形下，第一个因子对应孪生点类（具有相同邻域的顶点）的置换。我们还确定了$\mathbf{SIG}(\mathbb{V}_0)$的若干结构参数，包括孪生点类、连通性、控制数、直径、顶点度数与边数。
英文摘要
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 $Aut(\mathbf{SIG}(\mathbb V_0)) \cong \left( \prod_{i=1}^{q+1}S_q\times \prod_{i=1}^{\frac{q(q+1)}{2}}S_q \right) \rtimes S_{q+1}.$
Comments25 pages Revised version. The incorrect Theorem 2.5 has been removed and the corresponding result has been reformulated and proved independently. The manuscript has also been carefully revised for mathematical accuracy and clarity
