  We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of
distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for
every graph $G$ excluding four exceptions, we conjecture that $q(G)+q(G^c)\le
|G|+2$, where $G^c$ is the complement of $G$. We compute $q(G^c)$ for all trees
and all graphs $G$ with $q(G)=|G|-1$, and hence we verify the conjecture for
trees, unicyclic graphs, graphs with $q(G)\le 4$, and for graphs with $|G|\le
7$.
