  In this paper we completely characterize the graphs which have an edge
weighted adjacency matrix belonging to the class of $n \times n$ involutions
with spectrum equal to $\{ \lambda_1^{n-2}, \lambda_2^{2} \}$ for some
$\lambda_1$ and some $\lambda_2$. The connected graphs turn out to be the
cographs constructed as the join of at least two unions of pairs of complete
graphs, and possibly joined with one other complete graph.
