The last subconstituent of a bipartite \(Q\)-polynomial distance-regular graph
Let \(\Gamma\) be a bipartite distance-regular graph with diameter \(D\geq 3\). We define the depth \(d\) of \(\Gamma\) by \(2d=\text{ max}\{i+j-D\;|\;0\leq i,j\leq D\) and \(p_{ij}^D\neq 0\}\). Lemma 8.3. Let \(\Gamma\) be a bipartite \(Q\)-polynomial distance-regular graph with diameter \(D\geq 3\) and depth \(d\). Fix integers \(i,j\) (\(0\leq i,j\leq D\)) such that \(i+j+D\) is even and such that \(D\leq i+j\leq D+2d\). Then \(p_{ij}^D\neq 0\). Let \(\Gamma\) be a bipartite \(Q\)-polynomial distance-regular graph with diameter \(D\geq 3\) and depth \(d\). Fix any vertex \(x\) and let \(\Gamma_D\) denote the set of vertices at distance \(D\) from \(x\). Then (Theorems 9.2 and 9.6) for \(y,z\in \Gamma_D\), \(d_\Gamma(y,z)=2d_{\Gamma_D^2}(y,z)\) where \(\Gamma_D^2\) is the \(Q\)-polynomial distance-regular graph of diameter \(d\) with the following intersection numbers: \[ c_i^{(D)}=\frac{c_{2i}e_{d-1,2i-1}}{c_2},\quad b_i^{(D)}=\frac{b_{2i}n_{d-1,2i+1}}{c_2},\quad \text{ where} \] \[ n_{ij}=b_i\frac{(\theta_0^*-\theta_i^*)(\theta_{D-1}^*-\theta_{j-1}^*) -(\theta_1^*-\theta_{i+1}^*)(\theta_D^*-\theta_j^*)} {(\theta_0^*-\theta_i^*)(\theta_{j+1}^*-\theta_{j-1}^*)}, \] \[ e_{ij}=b_i\frac{(\theta_1^*-\theta_{i+1}^*)(\theta_D^*-\theta_j^*) -(\theta_0^*-\theta_i^*)(\theta_{D-1}^*-\theta_{j+1}^*)} {(\theta_0^*-\theta_i^*)(\theta_{j+1}^*-\theta_{j-1}^*)}. \]
scientific article; zbMATH DE number 1957109