A new feasibility condition for distance-regular graphs (Q1084413)

The main result, described in the title, is the following: Let \(\Gamma\) be a distance-regular graph with valency k (\(\geq 2)\) and diameter \(\geq 2\). Let \(\lambda_ 1\) and \(\lambda_{\min}\) be the second largest and the least eigenvalues of \(\Gamma\) and suppose that the multiplicity m(\(\lambda)\) of some eigenvalue \(\lambda\) (\(\neq k)\) satisfies \(m(\lambda)<k\). Then \(\lambda =\lambda_ 1\) or \(\lambda_{\min}\) and either (i) \(\lambda\) is an integer such that \(1+\lambda\) divides the intersection number \(b_ 1\), or (ii) \(\lambda\) and \(b_ 1/(1+\lambda)\) are algebraic integers, \(m(\lambda_ 1)=m(\lambda_{\min})\), and \(\lambda_ 1\) and \(\lambda_{\min}\) are algebraic conjugates over Q.