Maximality of the signless Laplacian energy

We study the problem of determining the graph with n vertices having largest signless Laplacian energy. We conjecture it is the complete split graph whose independent set has (roughly) 2n∕3 vertices. We show that the conjecture is true for several classes of graphs. In particular, the conjecture holds for the set of all complete split graphs of order n, for trees, for unicyclic and bicyclic graphs. We also give conditions on the number of edges, number of cycles and number of small eigenvalues so the graph satisfies the conjecture.