Characterization of digraphs with three complementarity eigenvalues.

Spectral graph theory probes the relation among structural graph attributes and the spectral decomposition of some matrices associated with the graph. One of the tasks considered in this area was the characterization of a graph by its spectrum. Lately, the concept of complementarity eigenvalues for matrices has become the central point of attraction. It has been noted that this complementarity spectrum permits distinguishing more graphs than the usual eigenvalues, so a question that arises naturally is whether a graph is found by its complementarity eigenvalues. That is, whether only connected isomorphic graphs share their complementarity spectra. This question remains open as of today. It is worth pointing out that there exist examples of non-isomorphic digraphs with the same complementarity spectrum. But, several questions concerning digraph characterization through the complementarity spectrum remain unanswered. The authors of this paper address the problem of characterizing all the strongly connected digraphs with exactly three complementarity eigenvalues.