Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\bar{G})=\mathrm{\mid }E(\bar{G})\mathrm{\mid }\le \lfloor n\mathrm{/}2\rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\bar{G})\le n-3$ satisfies $q(G)=2$. We show that this conjecture is true if $\bar{G}$ is bipartite and in other sporadic cases. Furthermore, we characterize $G$ with $\bar{G}$ bipartite and $e(\bar{G})=n-2$ for which $q(G)>2$.