ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\)

For a distance-regular graph \(\Gamma\) of diameter 3, the \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found parameters \(\Gamma_2\) given intersection array (independently, were by A.A. Makhnev D.V.Paduchikh). In this case, has an eigenvalue \(a_2-c_3\). paper, we study graphs with \(\theta=1\). particular, prove that, \(Q\)-polynomial from series arrays \(\{2c_3+a_1+1,2c_3,c_3+a_1-c_2;1,c_2,c_3\}\), equality \(c_3=4 (t^2+t)/(4t+4-c_2^2)\) holds. Moreover, \(t\le 100000\), there is unique feasible \(\{9,6,3;1,2,3\}\) corresponding to Hamming (or Doob) \(H(3,4)\). addition, parametrizations \(\theta_2=1\) \(\theta_3=a_2-c_3\).