Abstract
Motivated by conjectures of Karloff, and Van Dam and Sotirov on the smallest eigenvalues of Johnson and Hamming graphs, Brouwer, Cioabă, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, Bilinear forms graphs and Hermitian forms graphs. In this paper, we prove some of their conjectures.
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