  A regular clique in a regular graph is a clique such that every vertex
outside of the clique is adjacent to the same positive number of vertices
inside the clique. We continue the study of regular cliques in edge-regular
graphs initiated by A. Neumaier in the 1980s and attracting current interest.
We thus define a Neumaier graph to be an non-complete edge-regular graph
containing a regular clique, and a strictly Neumaier graph to be a non-strongly
regular Neumaier graph. We first prove some general results on Neumaier graphs
and their feasible parameter tuples. We then apply these results to determine
the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the
parameter tuples for all strictly Neumaier graphs having at most $24$ vertices.
Finally, we give two sequences of graphs, each with $i^{\text{th}}$ element a
strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a
positive integer) and having parameters of an affine polar graph as an
edge-regular graph. This answers questions recently posed by G. Greaves and J.
Koolen.
