On spectral integral variations of mixed graphs

In this paper, we characterize the mixed graphs with exactly one Laplacian eigenvalue moving up by an integer and other Laplacian eigenvalues remaining invariant when an edge is added. The results extend those of Fan [Linear and Multilinear Algebra 50 (2002) 133] for general graphs, and So [Linear and Multilinear Algebra 46 (1999) 193] for simple graphs.