On the Laplacian spread of graphs

The Laplacian spread s ( G ) of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G . Several upper bounds of Laplacian spread and corresponding extremal graphs are obtained in this paper. Particularly, if G is a connected graph with n ( ≥ 5 ) vertices and m ( n − 1 ≤ m ≤ n + 1 ) edges, then s ( G ) ≤ n − 1 with equality if and only if G is obtained from K 1 , n − 1 by adding m − n + 1 edges.