文摘
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 nn vertices having largest (signless) Laplacian energy for 3≤n≤133≤n≤13. For n≥11n≥11, we conjecture that the graph consisting of a triangle together with n−3n−3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on nn vertices. We prove this conjecture for many classes of graphs, depending on σσ, the number of (signless) Laplacian eigenvalues bigger than or equal to 2.