Line star sets for Laplacian eigenvalues

Let G be a mixed graph with a nonzero Laplacian eigenvalue μ of multiplicity k. A line star set for μ in G is a set Y of k edges of G such that μ is not a Laplacian eigenvalue of G−Y. It is shown that line star set exists for any nonzero Laplacian eigenvalue of any mixed graph. Some basic properties for line star sets are given. We also obtain some results on line star sets in undirected graphs.