The Laplacian spread of line graphs

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of its Laplacian matrix. In this paper, we determine the graphs whose line graphs achieve the smallest and the largest Laplacian spread, respectively, among all connected graphs with m≥3 edges, and we also determine the unique tree whose line graph achieves the smallest Laplacian spread among all non-star trees with m≥3 edges.