Amply regular graphs and block designs

We study the amply regular diameter $d$ graphs $\Gamma$ such that for some vertex $a$ the set of vertices at distance $d$ from $a$ is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance $d-1$ from $a$. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex $a$ if and only if the graph is distance-regular and for each vertex $a$ the subgraph $\Gamma_3(a)$ is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.