Inverse Problems in the Theory of Distance-Regular Graphs: Dual 2-Designs

Let Gamma be a distance-regular graph of diameter 3 with a strongly regular graph G3. Finding the parameters of Gamma(3) from the intersection array of G is a direct problem, and finding the intersection array of Gamma from the parameters of Gamma(3) is its inverse. The direct and inverse problems were solved by A.A. Makhnev and M.S. Nirova: if a graph Gamma with intersection array {k, b(1), b2(;) 1, c(2), c(3)} has eigenvalue theta(2) = -1, then the graph complementary to G3 is pseudogeometric for pGc3 (k, b1/c2). Conversely, if G3 is a pseudo-geometric graph for pG(alpha)(k, t), then G has intersection array {k, c(2)t, k - alpha + 1; 1, c(2), alpha}, where k - alpha + 1 <= c(2)t <= k and 1 <= c(2) <= a. Distance-regular graphs Gamma of diameter 3 such that the graph Gamma(3) ((Gamma) over bar) is pseudogeometric for a net or a generalized quadrangle were studied earlier. In this paper, we study intersection arrays of distance-regular graphs Gamma of diameter 3 such that the graph Gamma(3) ((Gamma) over bar (3)) is pseudogeometric for a dual 2-design pGt+1(l, t). New infinite families of feasible intersection arrays are found: {m(m(2)- 1), m(2)(m- 1), m(2); 1, 1, (m(2)- 1)(m- 1)}, {m(m+ 1), (m+ 2)(m- 1), m+ 2; 1, 1, m(2)- 1}, and {2m(m- 1), (2m- 1)(m- 1), 2m- 1; 1, 1, 2(m- 1)(2)}, where m = +/- 1 (mod 3). The known families of Steiner 2-designs are unitals, designs corresponding to projective planes of even order containing a hyperoval, designs of points and lines of projective spaces PG(n, q), and designs of points and lines of affine spaces AG(n, q). We find feasible intersection arrays of a distanceregular graph G of diameter 3 such that the graph Gamma(3) ((Gamma) over bar (3)) is pseudogeometric for one of the known Steiner 2-designs.