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    On the signless laplacian and normalized signless laplacian spreads of graphs
    (Springer Heidelberg, 2023) Milovanovic, Emina; Altındağ, Şerife Burcu Bozkurt; Matejic, Marjan; Milovanovic, Igor
    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.