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    題名: A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS
    作者: Li, Hui Shu;Li, Hong Hai
    關鍵詞: signed graph;Laplacian;eigenvalues;balancedness
    日期: 2016-09
    上傳時間: 2017-01-17 09:33:54 (UTC+8)
    出版者: 淡江大學出版中心
    摘要: Let Γ=(G,σ)Γ=(G,σ) be a connected signed graph, and L(Γ)L(Γ) be its Laplacian and L(Γ)L(Γ) its normalized Laplacian with eigenvalues λ1≥λ2≥⋯≥λnλ1≥λ2≥⋯≥λn and μ1≥μ2≥⋯≥μnμ1≥μ2≥⋯≥μn, respectively. It is known that a signed graph ΓΓ is balanced if and only if λn=0λn=0 (or μn=0μn=0). We show that λnλn and μnμn measure how much ΓΓ is far from being balanced by proving that
    μn(Γ)λn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},≤min{λ1(Γ′):Γ−Γ′isbalanced},
    μn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},λn(Γ)≤min{λ1(Γ′):Γ−Γ′isbalanced},
    where ν(Γ)ν(Γ) (resp. ϵ(Γ)ϵ(Γ)) denotes the frustration number (resp. the frustration index) of ΓΓ, that is the minimum number of vertices (resp. edges) to be deleted such that the signed graph is balanced.
    關聯: Tamkang Journal of Mathematics 47(3), pp.271-278
    DOI: 10.5556/j.tkjm.47.2016.1942
    顯示於類別:[淡江數學期刊] 第47卷第3期

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