Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius Groups

Let $G=N⋊H$ be a finite Frobenius group with $\mathrm{\mid }N\mathrm{\mid }=q$ and $\mathrm{\mid }H\mathrm{\mid }=q-1$. 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 $\mathrm{AGL}(1,q)$ over every finite field.