The Laplacian energy of threshold graphs and majorization

We study the Laplacian energy of threshold graphs, inspired by the recent results of Vinagre, Del-Vecchio, Justo and Trevisan [22]. In particular, we compute the degree sequences of threshold graphs that maximize (or minimize) the Laplacian energy for a fixed number of vertices and edges. The analysis involves combinatorial methods using Ferrers diagrams and ideas from majorization theory. Some new inequalities for threshold degree sequences are obtained in this process. In the review process a referee pointed out that, recently and independently, Helmberg and Trevisan [14] obtained very similar results, and we discuss this connection.