Bipartite \(Q\)-polynomial distance-regular graphs (Q1889840)

Let \(\Gamma\) denote a bipartite \(Q\)-polynomial distance-regular graph with diameter \(D\geq 4\). Then the intersection numbers of \(\Gamma\) are determined by \(D\) and two real scalars \(q\) and \(s^*\). It is proved that \(s^*=0\) if \(D\geq 12\). Theorem 1.1. Let \(\Gamma\) be a bipartite distance-regular graph with diameter \(D\geq 12\). Then \(\Gamma\) is \(Q\)-polynomial if and only if the following conditions (i)--(iv) hold: (i) \(\Gamma\) is the ordinary \(2D\)-cycle. (ii) \(\Gamma\) is the Hamming cube \(H(D,2)\). (iii) \(\Gamma\) is the antipodal quotient of \(H(2D,2)\). (iv) The intersection numbers of \(\Gamma\) satisfy \[ c_i=\frac{q^i-1}{q-1},\;b_i=\frac{q^D-q^i}{q-1},\;(0\leq i\leq D), \] where \(q\) is an integer not less than 2. The intersection numbers given in the case (iv) above are realized by both the bipartite dual polar graphs and the Hemmeter graph.