发表机构
Institute of Mathematics, University of Warmia and Mazury in Olsztyn（奥尔什丁瓦尔米亚-马祖里大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了二维q元单纯形码诱导的Grassmann子图直径对所有素数幂q均为3（q=2,3例外），从而完全确定了该直径，方法结合桥引理、计数论证与矩计算。
AI 中文摘要
设$\Gs(2,q)$为Grassmann图中由$\F_q^{q+1}$的二维子空间构成的子图，其由$q$元二维单纯形码诱导。$q=2,3$的情形是初等的，而Kwiatkowski和Pankov确定了$q=4$时的完整距离关系，并特别证明了$\diam \Gs(2,4)=3$。我们证明对于每个素数幂$q\ge 5$，有$\diam \Gs(2,q)=3$。因此，$\Gs(2,q)$的直径对所有$q$完全确定：当$q=2$时为$0$，当$q=3$时为$2$，当$q\ge4$时为$3$。主要成分是一个基于Marshall Hall关于有限阿贝尔群定理的桥引理；它给出了统一的上界$3$。对于$q\ge7$，下界由计数论证得出。其余情形通过简短的矩计算处理；对于$q=5$，我们还给出了一个独立的乘积证明。
英文摘要
Let $Γ^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$Γ^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\geq5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.
