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    Title: A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS
    Authors: Li, Hui Shu;Li, Hong Hai
    Keywords: signed graph;Laplacian;eigenvalues;balancedness
    Date: 2016-09
    Issue Date: 2017-01-17 09:33:54 (UTC+8)
    Publisher: 淡江大學出版中心
    Abstract: 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.
    Relation: Tamkang Journal of Mathematics 47(3), pp.271-278
    DOI: 10.5556/j.tkjm.47.2016.1942
    Appears in Collections:[淡江數學期刊] 第47卷第3期

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