The Laplacian spectral radius of bicyclic graphs with a given girth

Let ℬ(n,g)ℬ(n,g) be the class of bicyclic graphs on nn vertices with girth gg. Let ℬ1(n,g)ℬ1(n,g) be the subclass of ℬ(n,g)ℬ(n,g) consisting of all bicyclic graphs with two edge-disjoint cycles and ℬ2(n,g)=ℬ(n,g)∖ℬ1(n,g)ℬ2(n,g)=ℬ(n,g)∖ℬ1(n,g). This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ℬ1(n,g)ℬ1(n,g) and ℬ2(n,g)ℬ2(n,g), respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for ℬ(n,g)ℬ(n,g) are also obtained.