For graphs on n vertices whose complement is bipartite with at most n-3 edges, two distinct eigenvalues are always achievable; at the n-2 edge threshold, the only q=3 case is a double-star plus an isolated vertex.
Sparsity of graphs that allow tw o distinct eigenvalues
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Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues
For graphs on n vertices whose complement is bipartite with at most n-3 edges, two distinct eigenvalues are always achievable; at the n-2 edge threshold, the only q=3 case is a double-star plus an isolated vertex.