The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

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