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 Gn,k, where Gn,k is obtained from the complete graph Kn-k by attaching paths of almost equal lengths to all vertices of Kn-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.