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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 smallesteigenvalue of (distance-$j$) Johnson graphs. More generally, we study thesmallest eigenvalue and the second largest eigenvalue in absolute value of thegraphs of the relations of classical $P$- and $Q$-polynomial associationschemes.

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