Abstract

The complementarity spectrum of a graph consists of a finite collection of different scalars associated with the spectral radii of their induced subgraphs. The separability index of a class of graphs is the minimal number of successive complementarity eigenvalues, starting from the largest one, that is needed to distinguish the graphs in the class. For any given n≥15, we exhibit a pair G and H of order n graphs whose n−13 largest complementarity eigenvalues are equal. As a consequence, we deduce that the separability index of the set of connected graphs of order n grows at least linearly with n.