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.