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

09/26/2017

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by   Andries E. Brouwer, et al.

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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 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.