Tails of bipartite distance-regular graphs
Let \(\Gamma \) denote a bipartite distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(\theta _{0}>\theta _{1}>\cdots >\theta _{D}\) denote the eigenvalues of \(\Gamma \) and let \(E_{0}, E_{1},\ldots ,E_{D}\) denote the associated primitive idempotents. Fix \(s\) \((1\leq s\leq D-1)\); \(E_{s}\) is said to be a tail whenever the entrywise product \(E_{s}\circ E_{s}\) is a linear combination of \(E_{0}\), \(E_{s}\) and at most one other primitive idempotent of \(\Gamma \). Let \(q_{ij}^{h}\) \((0\leq h,i,j\leq D)\) denote the Krein parameters of \(\Gamma \) and let \(\Delta \) denote the undirected graph with vertices \(0,1,\ldots ,D\) where two vertices \(i,j\) are adjacent whenever \(i\neq j\) and \(q_{ij}^{s}\neq 0\). The main result of this paper asserts that \(E_{s}\) is a tail if and only if one of (i)--(iii) holds: (i) \(\Delta \) is a path; (ii) \(\Delta \) has two connected components, each of them is a path, one consisting of the even vertices and one consisting of the odd vertices; (iii) \(D=6\) and \(\Delta \) has two connected components, one is a path on four even vertices and the other is a clique on three odd vertices.
scientific article; zbMATH DE number 1884183