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On the [formula omitted]th Laplacian eigenvalues of trees with perfect matchings

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
432
Issue
4
Identifiers
DOI: 10.1016/j.laa.2009.10.015
Keywords
  • Tree
  • Laplacian Eigenvalue
  • Perfect Matchings
  • Bound

Abstract

Abstract Let T n + be the set of all trees of order n with perfect matchings. In this paper, we prove that for any tree T ∈ T n + , its k th largest Laplacian eigenvalue μ k ( T ) satisfies μ k ( T ) = 2 when n = 2 k , and μ k ( T ) ⩽ ⌈ n 2 k ⌉ + 2 + ( ⌈ n 2 k ⌉ ) 2 + 4 2 when n ≠ 2 k . Moreover, this upper bound is sharp when n = 0 ( mod 2 k ) .

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