On extensions of strongly regular graphs with eigenvalue 4

J. Koolen posed the problem of studying distance regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue ${}\le t$ for a given positive integer $t$. This problem was solved earlier for $t=3$. A program of studying distance regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue $r$, $3< r\le 4$, was started by the first author in his preceding paper. In this paper, a reduction to local exceptional graphs is performed. In the present work we find parameters of exceptional strongly regular graphs with nonprincipal eigenvalue 4. In addition, we prove that a distance regular graph in which neighborhoods of vertices are exceptional nonpseudogeometric strongly regular graphs with nonprincipal eigenvalue 4 has degree at most 729.