  We characterise graphs that have three distinct eigenvalues and coherent
ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter
to specific quasi-symmetric 2-designs. This characterisation leads to the
discovery of a new biregular graph with three distinct eigenvalues.
Additionally, we demonstrate that the coherent rank of a triregular graph with
three distinct eigenvalues is at least 14. Finally, we introduce a
conjecturally infinite family of biregular graphs with three distinct
eigenvalues, obtained by switching the block graphs of orthogonal arrays.
