New Upper Bounds for the Laplacian Spectral Radius of Graphs

Let D(G) and A(G) be the degree diagonal matrix and the adjacency matrix of a graph G,respectively.The Laplacian matrix of G is defined as L(G)=D(G)-A(G).In this paper,two new upper bounds for the Laplacian spectral radius of G in terms of the edge number,the vertex number,the largest degree,the second largest degree and the smallest degree of G are given by applying non-negative matrix theory and graph theory.Moreover,all extremal graphs which have these upper bounds are determined.Finally,two examples are given to show that our upper bounds can obtain smaller estimation value for the Laplacian spectral radius,it means that our upper bounds improve the results in previous papers in a way.