Maximum Laplacian energy of unicyclic graphs

We study the Laplacian and the signless Laplacian energy of connected unicyclic graphs, obtaining a tight upper bound for this class of graphs. We also find the connected unicyclic graph on n vertices having largest (signless) Laplacian energy for 3≤n≤13. For n≥11, we conjecture that the graph consisting of a triangle together with n−3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on n vertices. We prove this conjecture for many classes of graphs, depending on σ, the number of (signless) Laplacian eigenvalues bigger than or equal to 2.