Main $\mathbf{Q}$-eigenvalues of quasi-threshold graphs

In this note, we present a structural description of certain connected cographs having $k\ge 2$ main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are quasi-threshold graphs with two main $\mathbf{Q}$-eigenvalues. In addition, we describe all the quasi-threshold graphs belonging to the subclass of generalized core-satellite graphs with $k\ge 2$ main $\mathbf{Q}$-eigenvalues.