The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

Abstract

\u3cp\u3eWe prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.\u3c/p\u3