A remark on association schemes with two P-polynomial structures
\textit{E. Bannai} and \textit{E. Bannai} [Eur. J. Comb. 1, 289-298 (1980; Zbl 0458.94047)] classified symmetric association schemes \((X,\{R_ 0,\ldots,R_ d\})\) such that for two different choices of \(i\), the graph \((X,R_ i)\) is distance-regular with diameter d. Using the standard notation \(a_ i = p^ i_{1,i}\), \(b_ i=p^ i_{1,i+1}\), \(c_ i= p^ i_{1,i-1}\), \(k_ i = p^0_{i,i}\), \(k = k_1\), \(\lambda =a_1\), \(\mu =c_2\) for the graph \(\Gamma = (X,R_1)\) and \(\mu_{\Delta} = c_2(\Delta)\) for the graph \(\Delta =(X,R_ i)\), the author proves the following theorem. Theorem. Let \(\Gamma\), \(\Delta\) be two \(P\)-polynomial structures on the same association scheme, with distance correspondence \[ \begin{matrix} \Gamma & {\vrule depth1ex height2.5ex} & 0 & 1 & 2 & 3 & 4 & \ldots & d-3 & d-2 & d-1 & d \\ \\ \Delta & {\vrule depth1ex height2.5ex} & 0 & d & 2 & d-2 & 4 & \ldots & d-3 & 3 & d-1 & 1 \end{matrix} \] and \(d\geq5\). Assume that \(k>2\). Then (i) \(k(b_{d-1}-1) = (k_ d-1)\mu\); (ii) if \(d\geq7\), then \(d\) is odd and \(\Gamma\) and \(\Delta\) are bipartite, (iii) if \(d\geq6\) or \(\Gamma\) is not bipartite, then \(\mu\mu_\Delta = (b_{d-1}-1)^ 2+1\), \(k-1 = (b_{d-1}-1)(b_{d-1}-1+\mu,\) \(k_ d-1 = (b_{d-1}-1) (b_{d-1}-1+\mu_\Delta)\), \(k = \mu(b_{d-1}-1 + \mu_\Delta)\), \(k_ d = \mu_\Delta(b_{d-1}-1+\mu),\) \(b_{d-1} = (\mu+\mu_\Delta-3).\)
scientific article; zbMATH DE number 4143995
scientific article