An improved bound for strongly regular graphs with smallest eigenvalue $-m$

In 1979, Neumaier gave a bound on $\lambda$ in terms of $m$ and $\mu$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs. We improve this result. We also indicate how our methods can be used to give an alternate derivation of Bruck’s Completion Theorem for orthogonal arrays.