  A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by
replacing each vertex of $G$ by a clique or a coclique, where vertices of $H$
coming from different vertices of $G$ are adjacent if and only if the original
vertices are adjacent in $G$. If $G$ has no more than three vertices, $H$ has
all but at most three adjacency eigenvalues equal to $0$ or $-1$. In this paper
we consider the converse problem, and determine the class $\cal G$ of all
graphs with at most three eigenvalues unequal to $0$ and $-1$. Ignoring
isolated vertices, we find that $\cal G$ consists of all mixed extensions of
graphs on at most three vertices together with some particular mixed extensions
of the paths $P_4$ and $P_5$.
