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

paper

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

arXiv:1709.09011

Abstract

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.

30 pages, accepted by JCTB