Several sharp upper bounds for the largest laplacian eigenvalue of a graph

Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G,respectively.In this paper,we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix DKDand UBU,respectively,where D is the degree diagonal matrix of G and U=diag(d,d,:uv∈E（G）). And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G.Moreover,we determine all extremal graphs which achieve these upper bounds.Finally, some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.