Neumaier graphs of coherent rank five

Abstract: We construct an infinite family of Neumaier graphs of coherent rank five, answering the existence question at the smallest possible coherent rank beyond the strongly regular case. For every prime power $q\geqslant7$ with $q\equiv3\pmod4$, set $n=q+1$. Each graph in our construction has precisely five distinct eigenvalues, Neumaier parameters [ \left( n(n-1)(n-3), \frac{n^2(n-3)}2, \frac{n(n^2-n-8)}4; \frac{(n-2)^2}{2}, (n-1)(n-3) \right), ] and its adjacency matrix lies in the Bose--Mesner algebra of the four-class association scheme of Holzmann, Kharaghani, and Suda. Paley Hadamard matrices and Desarguesian mutually orthogonal Latin squares yield an infinite family whose smallest member has parameters $(280,160,96;18,35)$.