Two sharp upper bounds for the Laplacian eigenvalues

In this paper, we first obtain a sharp upper bound for the eigenvalues of the adjacency matrix of the line graph of a graph. Then this result is used to present a sharp upper bound for the Laplacian eigenvalues. Another sharp upper bound is presented also. Moreover, we determine all extreme graphs which attain these upper bounds. In last, two examples illustrate that our results are, in some sense, best.