  Motivated by the linear time algorithm that locates the eigenvalues of a
cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq
-1,0. 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.
