For a distance-regular graph (Formula presented.) of diameter 4, the graph (Formula presented.) can be strongly regular. In this case, the graph (Formula presented.) is strongly regular and complementary to (Formula presented.). Finding the intersection array of Γ from the parameters of (Formula presented.) is an inverse problem. In the present paper, the inverse problem is solved in the case of an antipodal graph (Formula presented.) of diameter 4. In this case, (Formula presented.) and (Formula presented.) is a strongly regular graph without triangles. Further, (Formula presented.) is an (Formula presented.)-graph only in the case (Formula presented.) and (Formula presented.). Earlier the authors proved that an (Formula presented.)-graph does not exist. A Krein graph is a strongly regular graph without triangles for which the equality in the Krein bound is attained (equivalently, (Formula presented.)). A Krein graph Krer with the second eigenvalue r has parameters (Formula presented.). For the graph Krer, the antineighborhood of a vertex is strongly regular with parameters (Formula presented.) and the intersection of the antineighborhoods of two adjacent vertices is strongly regularly with parameters. Let Γ be an antipodal graph of diameter 4, and let (Formula presented.) be a strongly regular graph without triangles. In this paper it is proved that (Formula presented.) cannot be a graph with parameters (Formula presented.), and if (Formula presented.) is a graph with parameters (Formula presented.), then (Formula presented.). It is proved that a distance-regular graph with intersection array (Formula presented.) exists only for r3, and, for a graph with array 967532r1r1132r7596, we have r2. © Pleiades Publishing, Ltd. 2022.