ON THE SIGNLESS LAPLACIAN AND NORMALIZED SIGNLESS LAPLACIAN SPREADS OF GRAPHS

Let G = (V, E), V = {v(1), v(2), ... , v(n)}, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d(1) >= d(2) >= ... >= d(n). 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 L+ = D-1/2L+D-1/2. The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as r(G) = gamma(+)(2)/gamma(+ )(n)and l(G) = gamma(+)(2) - gamma(+)(n), where gamma(+ )(1)>= gamma(+)(2) >=.. . >= gamma(+)(n) >= 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.