The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth - Li, Ke; Wang, Ligong; Zhao, Guopeng (The Electronic Journal of Combinatorics, 2011)
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 B_1(n,g) be the subclass of B(n,g) consisting of all bicyclic graphs with two edge-disjoint cycles and B_2(n,g)=B(n,g)B_1(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.