Inverse problems in the class of distance-regular graphs of diameter 4

For a distance-regular graph of diameter 4, the graph Delta = Gamma(1,2) can be strongly regular. In this case, the graph Gamma(3,4) is strongly regular and complementary to Delta. Finding the intersection array of from the parameters of Gamma(3,4) is an inverse problem. In the present paper, the inverse problem is solved in the case of an antipodal graph of diameter 4. In this case, r = 2 and Gamma(3,4) is a strongly regular graph without triangles. Further, Gamma is an AT4(p, q, r)-graph only in the case q = p+2 and r = 2. Earlier the authors proved that an AT4(p, p+2, 2)-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, q(22)(2) = 0). A Krein graph Kre(r) with the second eigenvalue r has parameters ((r(2) + 3r)(2), r(3) + 3r(2) + r, 0, r(2) + r). For the graph Kre(r), the antineighborhood of a vertex is strongly regular with parameters ((r(2) + 2r - 1)(r(2) + 3r + 1), r(3) + 2r(2), 0, r(2)) and the intersection of the antineighborhoods of two adjacent vertices is strongly regularly with parameters ((r(2) + 2r)(r(2) + 2r - 1), r(3) + r(2) - r, 0, r(2) - r). Let Gamma be an antipodal graph of diameter 4, and let Delta = Gamma(3,4) be a strongly regular graph without triangles. In this paper it is proved that Delta cannot be a graph with parameters ((r(2) + 2r - 1)(r(2) + 3r + 1), r(3) + 2r(2), 0, r(2)), and, if Delta is a graph with parameters ((r(2) + 2r)(r(2) + 2r - 1), r(3) + r(2) - r, 0, r(2) - r), then r > 3. It is proved that a distance-regular graph with intersection array {32, 27, 12(r - 1)/r, 1; 1, 12/r, 27, 32} exists only for r = 3, and, for a graph with array {96, 75, 32(r - 1)/r, 1; 1, 32/r, 75, 96}, we have r = 2.