Some sharp upper bounds of Laplacian spectral radius of trees in terms of order,diameter,pendant vertex number,covering number,edge covering number or total independence number are given.And the ninth to thirteenth largest values of Laplacian spectral radius over the class of trees on a given order are also given.
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.
This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result （Theorem 13）, Grone and Merris＇ conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221-229 （1994）] is partially proved.
