Inverse Problems in the Class of Distance-Regular Graphs of Diameter $$4$$

For a distance-regular graph $$\Gamma$$ 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 $$\Gamma$$ 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 $$\Gamma$$ 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, $$q^{2}_{22}=0$$ ). A Krein graph $$\mathrm{Kre}(r)$$ with the second eigenvalue $$r$$ has parameters $$((r^{2}+3r)^{2},r^{3}+3r^{2}+r,0,r^{2}+r)$$ . For the graph $$\mathrm{Kre}(r)$$ , 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 $$r=2$$ .