On the signless Laplacian and normalized signless Laplacian spreads of graphs

Abstract

Let G = (V, E), V = {v₁, v₂, …, v_n}, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d₁ ≽ d₂ ≽ … ≽ 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 $r\left( G \right) = \gamma _2^ + /\gamma _n^ +$. The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as $l\left( G \right) = \gamma _2^ + - \gamma _n^ +$, where $\gamma _1^ + \geqslant \gamma _2^ + \geqslant \ldots \geqslant \gamma _n^ + \geqslant 0$ are eigenvalues of ${{\cal L}^ + }$. 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.