On extensions of exceptional strongly regular graphs with eigenvalue 3

The study of distance regular graphs in which neighborhoods of vertices are strongly regular graphs with eigenvalue 3 was initiated in Makhnev's previous works. In particular, he reduced these graphs to graphs in which neighborhoods of vertices are exceptional graphs or pseudogeometric graphs for $pG_{s-3}(s,t)$. Makhnev and Paduchikh found parameters of exceptional graphs (see the Proposition). In the present paper, we study amply regular graphs in which neighborhoods of vertices are exceptional strongly regular graphs with eigenvalue 3 and parameters from conditions 3–6 of the Proposition.