Multiplicity of eigenvalues of cographs

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G (Jacobs et al., 2017), we investigate the multiplicity of eigenvalue λ for λ≠0,−1. For cographs with balanced cotrees we determine explicitly the highest value for the multiplicity. The energy of a graph is defined as the sum of absolute values of the eigenvalues. A graph G on n vertices is said to be borderenergetic if its energy equals the energy of the complete graph Kn. We present families of non-cospectral and borderenergetic cographs.