In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G - v be a graph obtained from graph G by deleting its vertex v and kappa(i)(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that kappa(i+1) (G) - 1 <= kappa(i)(G - v) <= kappa(i)(G). Next, we consider the third largest eigenvalue kappa(3) (G) and we give a lower bound in terms of the third largest degree d(3) of the graph G. In particular, we prove that kappa(3)(G) >= d(3)(G) - root 2. Furthermore, we show that in several situations the latter bound can be increased to d(3) - 1. (C) 2011 Elsevier Inc. All rights reserved.

A note on the signless Laplacian eigenvalues of graphs

BELARDO, FRANCESCO
2011

Abstract

In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G - v be a graph obtained from graph G by deleting its vertex v and kappa(i)(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that kappa(i+1) (G) - 1 <= kappa(i)(G - v) <= kappa(i)(G). Next, we consider the third largest eigenvalue kappa(3) (G) and we give a lower bound in terms of the third largest degree d(3) of the graph G. In particular, we prove that kappa(3)(G) >= d(3)(G) - root 2. Furthermore, we show that in several situations the latter bound can be increased to d(3) - 1. (C) 2011 Elsevier Inc. All rights reserved.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/1914452
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