  A Neumaier graph is a non-complete edge-regular graph containing a regular
clique. A Neumaier graph that is not strongly regular is called a strictly
Neumaier graph. In this work we present a new construction of strictly Neumaier
graphs, and using Jacobi sums, we show that our construction produces
infinitely many instances. Moreover, we prove some necessary conditions for the
existence of (strictly) Neumaier graphs that allow us to show that several
parameter sets are not admissible.
