On the Laplacian spectral radius of bipartite graphs with fixed order and size

Let Gn,m be the set of all connected bipartite graphs of order n and size m. In this paper, the problem on maximum Laplacian spectral radius of graphs in Gn,m is considered. Among Gn,m with n⩽m⩽2n−5, the largest Laplacian spectral radius of graphs is determined. As well the upper bound on Laplacian spectral radius of graphs among Gn,l(n−l) with 2⩽l⩽⌊n2⌋ is determined. All the corresponding extremal graphs are characterized, respectively.