On the multiplicity of eigenvalues of distance-regular graphs
The main goal of this note is to give better lower bounds on the multiplicity of eigenvalues in a distance-regular graph. The main result is Theorem 4: Let \(\Gamma\) be a distance-regular graph with \(a_d= 0\). Let \(\Theta\) be an eigenvalue of \(\Gamma\) with multiplicity \(f\). Suppose that the standard array belonging to \(\Theta\) satisfies \(u_2\neq u_0\neq u_d\). Then \(f> k+ b_{d- 1}- 1\) or \(\Theta\in \{\Theta_1, \Theta_d\}\) and either \(\Theta+ 1\) is an integer dividing \(b_1\) or \(\Theta_1+ 1\) and \(\Theta_d+ 1\) are conjugate algebraic integers dividing \(b_1\). This rules out the infinite series of feasible intersection arrays \[ \{\mu(2\mu+ 1), (\mu- 1)(2\mu+ 1), \mu^2, \mu; 1, \mu, \mu(\mu- 1), \mu(2\mu+ 1): \mu\geq 2\}. \]
scientific article; zbMATH DE number 803742