The Laplacian eigenvalues of mixed graphs Original Research Article

In this paper, we firstly give an upper bound for the second smallest Laplacian eigenvalue of mixed graphs which generalizes the results of Fiedler [Czechoslovake Math. J. 23 (1973) 298]. Then we present two sharp upper bounds for the largest Laplacian eigenvalues of mixed graphs in term of the largest, smallest degrees and average 2-degrees, which improve and generalize the main results of Merris [Linear Algebra Appl. 285 (1998) 33], Li and Pan [Linear Algebra Appl. 328 (2001) 153], respectively. In addition, we characterize all extreme mixed graphs which attain those upper bounds.