  We examine the capacity of the complementarity spectrum to distinguish
non-isomorphic digraphs. We focus on the seven families with exactly three
complementarity eigenvalues. Our findings reveal that in some, but not all
families, any two non-isomorphic members have different complementarity
spectrum. Complementarity eigenvalues outperform traditional eigenvalues in the
task of identifying graphs. Indeed, the question of whether graphs are uniquely
determined by their complementarity spectrum remains unresolved, highlighting
the significance of this tool in graph theory. Moreover, since the
complementarity spectrum of a digraph was characterized as the set of spectral
radii of the induced strongly connected subdigraphs, the results of this study
provides useful structural information for important families of digraphs.
