Exploring symmetries in cographs: Obtaining spectra and energies

We explore symmetries present in the cotree of a cograph G and apply these ideas to obtain parts of the spectra of matrices associated with G. In each case, the remaining eigenvalues may be obtained from the spectrum of a smaller matrix, thus reducing the complexity for the computation of the spectra of those matrices. The algorithm we introduce may be applied to any matrix whose underlying graph is a cograph, in a unified way. As applications, we study the energies of known matrices, such as the adjacency, Laplacian, signless Laplacian, distance and its Laplacians, and normalized Laplacian. We find infinite families of equienergetic cographs and discuss what integers can be energies of cographs.