Maximum Laplacian energy among threshold graphs

We impose some restrictions on the spectra of a threshold graph, allowing us to compute its Laplacian energy. We then show that the pineapple with clique number has largest Laplacian energy among all pineapples with n vertices and among all the graphs satisfying those conditions. We also find large families of threshold graphs that satisfy those restrictions and exhibit pairs of such graphs with the same Laplacian energy and different Laplacian spectra. We show by computation that the above pineapple has maximum Laplacian energy among all connected threshold graphs for all n<35.