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. © 2022, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.