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

Abstract

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.

Disclosure

“igid because N \ {1} is a single conjugacy class. For larger ratios it splits into several classes, so one should expect a matrix-valued or multi-fiber analogue rather than a single lexicographic blow-up. Declaration of generative AI use Generative AI tools were used during manuscript preparation to assist with drafting portions of the exposition and with mathematical reasoning. All arguments and conclusions were independently checked by the authors, who take full responsibility for the”

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2026-08-22T18:12:31.807729Z

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