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

Andries E. Brouwer, Sebastian M. Cioabă, Ferdinand Ihringer, Matt McGinnis

Published 2017-09-26Version 1

We prove a conjecture by Van Dam and 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.