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

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