On the signless Laplacian and normalized signless Laplacian spreads of graphs

Let G = (V, E), V = {v1, v2, …, vn}, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d1 ≽ d2 ≽ … ≽ dn. Denote by A and D the adjacency matrix and diagonal vertex degree matrix of G, respectively. The signless Laplacian of G is defined as L+ = D + A and the normalized signless Laplacian matrix as r(G)=γ2+/γn+. The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as l(G)=γ2+−γn+, where γ1+⩾γ2+⩾..⩾γn+⩾0 are eigenvalues of ℒ+. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.