On Distance-Regular Graphs of Diameter 3 with Eigenvalue Θ = 1

Abstract

For a distance-regular graph Γ of diameter 3, the graph Γi can be strongly regular for i = 2 or 3. J. Kulen and co-authors found the parameters of a strongly regular graph Γ2 given the intersection array of the graph Γ (independently, the parameters were found by A.A. Makhnev and D.V. Paduchikh). In this case, Γ has an eigenvalue a2 - c3. In this paper, we study graphs Γ with strongly regular graph Γ2 and eigenvalue θ = 1. In particular, we prove that, for a Q-polynomial graph from a series of graphs with intersection arrays {2c3 + a1 + 1, 2c3,c3 + a1 - c2; 1,c2,c3}, the equality c3 = 4(t2 + t)∕(4t + 4 - c22) holds. Moreover, for t ≤ 100000, there is a unique feasible intersection array {9, 6, 3; 1, 2, 3} corresponding to the Hamming (or Doob) graph H(3, 4). In addition, we found parametrizations of intersection arrays of graphs with θ2 = 1 and θ3 = a2 - c3.