Existence of Neumaier graphs of coherent rank five

The problem asks whether Neumaier graphs can have coherent rank $5$, the first possible rank beyond the strongly regular case. A new construction claims such graphs exist for every prime power $q \ge 7$ with $q \equiv 3 \pmod{4}$, using Paley Hadamard matrices and Desarguesian mutually orthogonal Latin squares.

Known results

• Nontrivial strongly regular graphs have coherent rank $3$; every Neumaier graph of coherent rank at most $4$ is strongly regular, leaving rank $5$ as the first unresolved case beyond it.
• A 2025 cyclotomic construction gives an infinite family with coherent rank $6$, but does not settle rank $5$.

September 2, 2026 construction claim

The reported construction claims an infinite family with coherent rank $5$ and exactly five distinct eigenvalues, including parameters $(280,160,96;18,35)$ for the smallest listed example. It establishes existence if verified, but the supplied evidence records no independent checking or classification.

Current status (as of September 2026): An arXiv construction claims existence of infinitely many coherent-rank-$5$ Neumaier graphs, while independent verification is unrecorded and classification remains open.