A bound on the Laplacian spread which is tight for strongly regular graphs

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. For Laplacian matrices of graphs, we find their upper bounds of largest eigenvalues, lower bounds of second-smallest eigenvalues and upper bounds of Laplacian spreads. The strongly regular graphs attain all the above bounds. Some other extremal graphs are also provided.