On strongly regular graphs with eigenvalue $\mu$ and their extensions

Let $\mathcal M$ be a class of strongly regular graphs for which $\mu$ is a non-principal eigenvalue. Note that the neighborhood of any vertex of an $AT4$ graph lies in $\mathcal M$. We describe parameters of graphs from $\mathcal M$ and find intersection arrays of $AT4$ graphs in which neighborhoods of vertices lie in chosen subclasses from $\mathcal M$. In particular, an $AT4$ graph in which the neighborhoods of vertices do not contain triangles is the Conway–Smith graph with parameters $(p,q,r)=(1,2,3)$ or the first Soicher graph with parameters $(p,q,r)=(2,4,3)$.