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+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\left( G \right) = \gamma _2^ + /\gamma _n^ + $$\end{document}. The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as l(G)=γ2+−γn+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\left( G \right) = \gamma _2^ + - \gamma _n^ + $$\end{document}, where γ1+⩾γ2+⩾...⩾γn+⩾0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^ + \geqslant \gamma _2^ + \geqslant \ldots \geqslant \gamma _n^ + \geqslant 0$$\end{document} are eigenvalues of ℒ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal L}^ + }$$\end{document}. 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.