On the Laplacian spread of graphs

The Laplacian spread s(G)s(G) of a graph GG is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of GG. Several upper bounds of Laplacian spread and corresponding extremal graphs are obtained in this paper. Particularly, if GG is a connected graph with nn(≥5)(≥5) vertices and mm(n−1≤m≤n+1)(n−1≤m≤n+1) edges, then s(G)≤n−1s(G)≤n−1 with equality if and only if GG is obtained from K1,n−1K1,n−1 by adding m−n+1m−n+1 edges.