A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS

淡江大學機構典藏 > 淡江出版期刊 > 淡江數學期刊 > 第47卷第3期 > Item 987654321/109351

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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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