A mathematical preprint gives an exact classification of Ramanujan Cayley graphs whose connection sets are normal, meaning they are unions of conjugacy classes. The result concerns finite Frobenius groups with a kernel N of size q and a complement H of size q−1.
The Ramanujan test is spectral. The analysis starts with arbitrary connected regular base graphs Y on q−1 vertices, using a least-eigenvalue cutoff greater than −2 for non-bipartite bases and a trace-of-the-square calculation for bipartite ones.
The structure behind the list
Every normal inverse-closed set S—one that contains an element whenever it contains its inverse—has a unique form determined by a binary choice, ε = 0 or 1, and a normal inverse-closed subset E of H. The set S generates G exactly when E generates H.
Writing Y as the Cayley graph Cay(H,E), the full graph becomes a lexicographic blow-up with uniform fibers. Between two fibers, adjacency is all-or-none; inside each fiber, there are no edges when ε = 0 and all possible edges when ε = 1.
The few base graphs that qualify
For one blow-up form, the permitted base patterns are the complete graph K_{q−1}; a complete bipartite graph K_{r,r} with q−1 = 2r; and K_{r,r} with a perfect matching removed, with q−1 = 2r and q ≥ 7.
For the other blow-up form, the only bases are K_{q−1} and the cocktail-party graph, meaning the complete graph K_{2r} with a perfect matching removed. The latter requires q−1 = 2r and q ≥ 5.
The group-level classification
Translating those base cases back to the group gives an if-and-only-if list of normal connection sets. Two entries are G with the identity removed and G with the kernel N removed.
The remaining entries are conditional. One takes every nonidentity element of N and adds the inverse image of H with the identity and a central involution removed; it requires q ≥ 5. Another is the inverse image of H with a normal subgroup K of index two removed. A final one removes a central involution outside K from that construction and requires q ≥ 7.
A separate count for the affine family
In the affine family AGL(1,q), the normal Ramanujan connection sets fall into two types when q is even, four types when q is congruent to 1 modulo 4, and five types when q is congruent to 3 modulo 4 and q ≥ 7.
The smallest case is handled separately: at q = 3, the paper identifies two distinct graphs, K6 and K3,3.
Where the theorem stops
The classification is for ratio-one Frobenius groups and normal, inverse-closed connection sets, so non-normal connection sets lie outside the result.
Publication note
The manuscript is an arXiv version 1 preprint dated 20 Aug 2026. Its authors disclose using generative AI for drafting and mathematical reasoning, and say they independently checked the arguments and conclusions and take responsibility for them.
Paper data and sources
Original title: Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius Groups
Authors: Ming-Hsuan Kang, Chi-Jung Yang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text