Maximal signless Laplacian spectral radius of bicyclic graphs with given independence number

Let B(n,α)be the class of bicyclic graphs on n vertices with independence numberα.Let B_1(n,α)be the subclass of B(n,α)consisting of all bicyclic graphs with two edge-disjoint cycles and B_2(n,α)=B(n,α)\B_1(n,α).This paper determined the unique graph with the maximal signless Laplacian spectral radius among all graphs in B_1(n,α) and B_2(n,α),respectively.Furthermore,the upper bound of the signless Laplacian spectral radius and the extremal graph for B(n,α)were also obtained.