On bipartite \(Q\)-polynomial distance-regular graphs with \(c_{2}=1\)
Let \(\Gamma \) denote a bipartite \(Q\)-polynomial distance-regular graph with diameter greater than or equal to three, valency \(k\geq 3\) and intersection number \(c_{2}=1\). In this paper it is shown that \(\Gamma \) has a certain equitable partition of its vertex set which involves \(4d-4\) cells. This partition is used to show that the intersection numbers \(a_{i},b_{i},c_{i}\) of \(\Gamma \) satisfy the following divisibility conditions: \(c_{i+1}-1\) divides \(c_{i}(c_{i}-1)\) for \(2\leq i\leq d-1\), \(b_{i-1}-1\) divides \(b_{i}(b_{i}-1)\) for \(1\leq i\leq d-1\) and \(k-2\) divides \((c_{3}-1)(c_{3}-2)\). Using these divisibility conditions it is shown that there does not exist a bipartite \(Q\)-polynomial distance-regular graph with valency \(k\geq 3\), \(c_{2}=1\) and diameter \(d=4\).
scientific article; zbMATH DE number 5124971