具有正规连接集的拉马努金凯莱图在比例一弗罗贝尼乌斯群中的应用
Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius Groups
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中文总结 AI 辅助
本文对比例一弗罗贝尼乌斯群$G=N\rtimes H$上的正规连接集拉马努金凯莱图进行分类，利用膨胀现象将其转化为补群$H$上的图分类，最终得到适用于$\text{AGL}(1,q)$等的完整分类结果。
中文摘要 AI 辅助
设$G=N\rtimes H$为有限弗罗贝尼乌斯群，其中$|N|=q$，$|H|=q-1$。我们对$G$的所有正规连接集拉马努金凯莱图进行分类，此处正规连接集指共轭类的并集。群论输入为一种简单的膨胀现象：每个此类凯莱图要么是$Y[\br{K_q}]$，要么是$Y[K_q]$，其中$Y$是补群$H$上的连通正则凯莱图。我们首先证明一个图论结果，对当$Y$是$q-1$个顶点上的任意连通正则图时的这两种形式的所有拉马努金图进行分类，证明结合了最小特征值大于$-2$的正则图的经典刻画，以及二分图情形下的二阶矩恒等式。将得到的五种图类型转换回$G$，可得到所有比例一弗罗贝尼乌斯群的完整分类，尤其适用于每个有限域上的$\text{AGL}(1,q)$。
英文摘要
Let $G=N\rtimes H$ be a finite Frobenius group with $|N|=q$ and $|H|=q-1$. We classify all Ramanujan Cayley graphs of $G$ whose connection sets are normal, in the sense of being unions of conjugacy classes. The group-theoretic input is a simple blow-up phenomenon: every such Cayley graph is either $Y[\overline{K_q}]$ or $Y[K_q]$ for a connected regular Cayley graph $Y$ on the complement $H$. We first prove a graph-theoretic result classifying all Ramanujan graphs of these two forms when $Y$ is an arbitrary connected regular graph on $q-1$ vertices. The proof combines the classical characterization of regular graphs with least eigenvalue greater than $-2$ with a second-moment identity in the bipartite case. Translating the resulting five graph types back to $G$ yields a complete classification for all ratio-one Frobenius groups, and in particular for $\operatorname{AGL}(1,q)$ over every finite field.
