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
2011-01-01
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.Pubblicazioni consigliate
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