On the spectral radius of weighted trees with fixed diameter and weight set

The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. We first give some perturbational results on the spectral radius of weighted graphs when some weights of edges are modified, then we derive the weighted tree with the largest spectral radius in the set of all weighted trees with fixed diameter and weight set. Furthermore, an open problem of spectral radius on weighted paths is solved [H.Z. Yang, G.Z. Hu, Y. Hong, Bounds of spectral radii of weighted tree, Tsinghua Science and Technology 8 (2003) 517–520].