The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. In this paper, we derive the sharp upper bound of spectral radius of all weighted trees on given order and edge independence number, and obtain all such trees that their spectral radius reach the upper bound.
The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix.Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979.In this paper,we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.
