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A note on the signless Laplacian eigenvalues of graphs

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
435
Issue
10
Identifiers
DOI: 10.1016/j.laa.2011.04.004
Keywords
  • Signless Laplacian
  • Interlacing Theorem
  • Equitable Partition
  • Third Largest Eigenvalue
Disciplines
  • Mathematics

Abstract

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 κ i ( G ) be the ith largest eigenvalue of the signless Laplacian of G, we show that κ i + 1 ( G ) - 1 ⩽ κ i ( G - v ) ⩽ κ i ( G ) . Next, we consider the third largest eigenvalue κ 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 κ 3 ( G ) ⩾ d 3 ( G ) - 2 . Furthermore, we show that in several situations the latter bound can be increased to d 3 - 1 .

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