An inequality involving the local eigenvalues of a distance-regular graph
Let \(\Gamma\) be a distance-regular graph with diameter \(D\geq 3\) and valency \(k\). Let \(X\) denote the vertex set of \(\Gamma\), and let \(x\in X\) and \(\Delta=\Gamma(x)\). Let \(T=T(x)\) denote the Terwilliger algebra of \(\Gamma\) with respect to \(x\). An irreducible \(T\)-module \(W\) is thin whenever dim \(E_i^*W\leq 1\) for \(0\leq i\leq D\). The endpoint of \(W\) is min\(\{i\;|\;E_i^*W\neq 0\}\). Let \(\eta_1\geq \eta_2\geq \cdots \geq \eta_k\) denote the eigenvalues of \(\Delta\). Let \(\Phi\) denote the set of distinct scalars of \(\eta_2,\dots ,\eta_k\). For \(\eta \in \Phi\) let \(m_\eta\) denote the number of times \(\eta\) appears among \(\eta_2,\dots,\eta_k\) and \(\widetilde \eta=-1-b_1/(1+\eta)\). Let \(p_{-1},p_0,\dots,p_D\) denote the polynomials in \({\mathbb R}(\lambda)\) satisfying \(p_{-1}=0\), \(p_0=1\) and \(\lambda p_i=c_{i+1}p_{i+1}+(a_i-c_{i+1}+c_i)p_i+b_ip_{i-1}\) (\(0\leq i\leq D-1\)). Theorem 13.5. For \(1\leq i\leq D-1\) we have \[ 1+\sum_{\eta\in \Phi, \eta\neq -1}\frac {p_{i-1}(\widetilde \eta)} {p_i(\tilde \eta)(1+\tilde \eta)}m_\eta\leq \frac {k}{b_1}. \] Theorem 13.6. The following are equivalent. (i) Equality holds in Theorem 13.5 for \(1\leq i\leq D-1\). (ii) Equality holds in Theorem 13.5 for \(i=D-1\). (iii) Every irreducible \(T\)-module with endpoint 1 is thin.
scientific article; zbMATH DE number 2082001