On the signless Laplacian spectral radius of graphs with cut vertices

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph G_n,k, where G_n,k is obtained from the complete graph K_n-k by attaching paths of almost equal lengths to all vertices of K_n-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.