On the kth Laplacian eigenvalues of trees with perfect matchings

Jianxi Li; Wai Chee Shiu; An Chang

doi:10.1016/j.laa.2009.10.015

On the kth Laplacian eigenvalues of trees with perfect matchings

Jianxi Li^*
, Wai Chee Shiu
, An Chang

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

10 Citations (Scopus)

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 kth largest Laplacian eigenvalue μ_k (T) satisfies μ_k (T) = 2 when n = 2 k, and μ_k (T) ≤ frac(⌈ frac(n, 2 k) ⌉ + 2 + sqrt((⌈ frac(n, 2 k) ⌉)² + 4), 2) when n ≠ 2 k. Moreover, this upper bound is sharp when n = 0 (mod 2 k).

Original language	English
Pages (from-to)	1036-1041
Number of pages	6
Journal	Linear Algebra and Its Applications
Volume	432
Issue number	4
DOIs	https://doi.org/10.1016/j.laa.2009.10.015
Publication status	Published - 1 Feb 2010

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Bound
Laplacian eigenvalue
Perfect matchings
Tree

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title = "On the kth Laplacian eigenvalues of trees with perfect matchings",

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

keywords = "Bound, Laplacian eigenvalue, Perfect matchings, Tree",

author = "Jianxi Li and Shiu, \{Wai Chee\} and An Chang",

note = "Funding Information: Partially supported by GRF, Research Grant Council of Hong Kong and Faculty Research Grant; FRG, Hong Kong Baptist University; The Natural National Science Foundation of China (No. 10871046). ∗ Corresponding author.",

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doi = "10.1016/j.laa.2009.10.015",

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T1 - On the kth Laplacian eigenvalues of trees with perfect matchings

AU - Li, Jianxi

AU - Shiu, Wai Chee

AU - Chang, An

N1 - Funding Information: Partially supported by GRF, Research Grant Council of Hong Kong and Faculty Research Grant; FRG, Hong Kong Baptist University; The Natural National Science Foundation of China (No. 10871046). ∗ Corresponding author.

PY - 2010/2/1

Y1 - 2010/2/1

N2 - Let Tn+ be the set of all trees of order n with perfect matchings. In this paper, we prove that for any tree T ∈ Tn+, its kth largest Laplacian eigenvalue μk (T) satisfies μk (T) = 2 when n = 2 k, and μk (T) ≤ frac(⌈ frac(n, 2 k) ⌉ + 2 + sqrt((⌈ frac(n, 2 k) ⌉)2 + 4), 2) when n ≠ 2 k. Moreover, this upper bound is sharp when n = 0 (mod 2 k).

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

KW - Bound

KW - Laplacian eigenvalue

KW - Perfect matchings

KW - Tree

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DO - 10.1016/j.laa.2009.10.015

M3 - Journal article

AN - SCOPUS:71649104321

SN - 0024-3795

VL - 432

SP - 1036

EP - 1041

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 4

ER -

On the kth Laplacian eigenvalues of trees with perfect matchings

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