AI 中文总结
该研究针对矩阵环正则图$\text{Γ}_n(q)$，通过嵌入全图、利用拉普拉斯不等式与矩阵环特征计算，得到其解离数的奇偶敏感上下界，并确定最小情形$\text{diss}(\text{Γ}_2(2))=3$。
AI 中文摘要
设$\boldsymbol{\text{Γ}_n(q)}$为顶点是有限域$\boldsymbol{\text{F}_q}$上$n$阶矩阵环$\boldsymbol{\text{Mat}_n(\text{F}_q)}$中可逆矩阵的图，两个不同矩阵相邻当且仅当它们的和是奇异的。解离集是诱导子图最大度至多为1的顶点集。我们通过将$\boldsymbol{\text{Γ}_n(q)}$嵌入到所有$\boldsymbol{\text{Mat}_n(\text{F}_q})$构成的全图$\boldsymbol{T_n(q)}$作为诱导子图，研究$\boldsymbol{\text{Γ}_n(q)}$的解离数。$k$独立集的一般拉普拉斯不等式，结合矩阵环加法群的显式特征计算，给出了与奇偶性相关的上界：对于固定$n$，奇数$q$时所得界的阶至多为$\boldsymbol{q^{n^2-n+1}}$，偶数$q$时至多为$\boldsymbol{q^{n^2-2n+2}}$，特别地，$\boldsymbol{\text{diss}(\text{Γ}_n(q))\boldsymbol{\text{≤}q^{n^2-n+1}-1}}$。下界方面，扩域$\boldsymbol{\text{F}_{q^n}}$的正则表示给出$\boldsymbol{\text{diss}(\text{Γ}_n(q))\boldsymbol{\text{≥}q^n-1}}$。我们给出完整证明，包括所需矩阵特征和的自包含推导，并确定了最小情形：$\boldsymbol{\text{diss}(\text{Γ}_2(2))=3}$。
英文摘要
Let $Γ_n(q)$ be the graph whose vertices are the invertible matrices in $\Mat_n(\F_q)$, with two distinct matrices adjacent whenever their sum is singular. A dissociation set is a vertex set inducing a graph of maximum degree at most one. We study the dissociation number of $Γ_n(q)$ by embedding it as an induced subgraph of the total graph $T_n(q)$ on all of $\Mat_n(\F_q)$. A general Laplacian inequality for $k$-independent sets, together with an explicit character computation for the additive group of the matrix ring, gives parity-sensitive upper bounds. For fixed $n$, the resulting bound is of order at most $q^{n^2-n+1}$ for odd $q$ and at most $q^{n^2-2n+2}$ for even $q$. In particular, \[ \diss(Γ_n(q))\le q^{n^2-n+1}-1. \] In the other direction, the regular representation of the extension field $\F_{q^n}$ gives $\diss(Γ_n(q))\ge q^n-1$. We give complete proofs, including a self-contained derivation of the required matrix character sum, and determine the smallest case: $\diss(Γ_2(2))=3$.
Comments18 pages, comments are welcome
