Graphs with three eigenvalues from group divisible designs

Abstract: We construct biregular graphs with three distinct eigenvalues from semiregular group divisible designs. For every prime power $q$, we use the spectral structure of Muzychuk's Construction~A on $q+1$ copies of $\operatorname{AG}(2,q<sup>2)$ to produce suitable group divisible designs. The resulting biregular graphs have $2q^5+q^4+q^2$ vertices and distinct eigenvalues $q<sup>5+q<sup>3+q<sup>2$, $q<sup>2$, and $-q<sup>2$. Their coherent closures have rank $12$ and type $\left[\begin{smallmatrix}3&amp;2\2&amp;5\end{smallmatrix}\right]$. We also determine the four-class symmetric association schemes that are the coherent closures of the auxiliary graphs in this family. Computational searches provide further graphs on $81$ and $156$ vertices, using Muzychuk's Construction~B and a generalised quadrangle, respectively; their coherent closures have ranks $13$ and $10$. The $156$-vertex example has a strongly regular design as its coherent closure, of type $\left[\begin{smallmatrix}3&amp;2\2&amp;3\end{smallmatrix}\right]$, answering a question of the authors. We also realise a $126$-vertex example by switching a strongly regular graph.