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In this paper we completely characterize the graphs which have an edgeweighted adjacency matrix belonging to the class of $n \times n$ involutionswith 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 thecographs constructed as the join of at least two unions of pairs of completegraphs, and possibly joined with one other complete graph.

Graphs that have a weighted adjacency matrix with spectrum $\{\lambda_1^{n-2}, \lambda_1^2\}$