On the multiplicities of normalized Laplacian eigenvalues of graphs

Shaowei Sun; Kinkar Chandra Das

doi:10.1016/j.laa.2020.09.022

On the multiplicities of normalized Laplacian eigenvalues of graphs

Shaowei Sun, Kinkar Chandra Das

Department of Mathematics

Zhejiang University of Science and Technology

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ₁≥ρ₂≥⋯≥ρ_n−1≥ρ_n=0. Let m_G(ρ_i) (1≤i≤n) be the multiplicity of the normalized Laplacian eigenvalue ρ_i of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have m_G(ρ_i)=n−3 for some i. We also discuss about graphs with m_G(ρ_n−2)=n−k. Moreover, we completely characterize the connected graphs with m_G(ρ_n−1)=n−k for sufficiently large n.

Original language	English
Pages (from-to)	365-385
Number of pages	21
Journal	Linear Algebra and Its Applications
Volume	609
DOIs	https://doi.org/10.1016/j.laa.2020.09.022
State	Published - 15 Jan 2021

Keywords

Graph
Multiplicity of eigenvalues
Normalized Laplacian matrix
Ramsey number

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10.1016/j.laa.2020.09.022

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title = "On the multiplicities of normalized Laplacian eigenvalues of graphs",

abstract = "Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. Let mG(ρi) (1≤i≤n) be the multiplicity of the normalized Laplacian eigenvalue ρi of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have mG(ρi)=n−3 for some i. We also discuss about graphs with mG(ρn−2)=n−k. Moreover, we completely characterize the connected graphs with mG(ρn−1)=n−k for sufficiently large n.",

keywords = "Graph, Multiplicity of eigenvalues, Normalized Laplacian matrix, Ramsey number",

author = "Shaowei Sun and Das, \{Kinkar Chandra\}",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2021",

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doi = "10.1016/j.laa.2020.09.022",

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T1 - On the multiplicities of normalized Laplacian eigenvalues of graphs

AU - Sun, Shaowei

AU - Das, Kinkar Chandra

PY - 2021/1/15

Y1 - 2021/1/15

N2 - Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. Let mG(ρi) (1≤i≤n) be the multiplicity of the normalized Laplacian eigenvalue ρi of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have mG(ρi)=n−3 for some i. We also discuss about graphs with mG(ρn−2)=n−k. Moreover, we completely characterize the connected graphs with mG(ρn−1)=n−k for sufficiently large n.

AB - Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. Let mG(ρi) (1≤i≤n) be the multiplicity of the normalized Laplacian eigenvalue ρi of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have mG(ρi)=n−3 for some i. We also discuss about graphs with mG(ρn−2)=n−k. Moreover, we completely characterize the connected graphs with mG(ρn−1)=n−k for sufficiently large n.

KW - Graph

KW - Multiplicity of eigenvalues

KW - Normalized Laplacian matrix

KW - Ramsey number

UR - https://www.scopus.com/pages/publications/85091648177

U2 - 10.1016/j.laa.2020.09.022

DO - 10.1016/j.laa.2020.09.022

M3 - Article

AN - SCOPUS:85091648177

SN - 0024-3795

VL - 609

SP - 365

EP - 385

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

On the multiplicities of normalized Laplacian eigenvalues of graphs

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Keywords

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