Lower bounds of the Laplacian graph eigenvalues |
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Authors: | Aleksandar Torgaev Miroslav Petrovi |
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Institution: | aMathematical Faculty, Studentski trg 16a, 11000 Belgrade, Serbia and Montenegro;bFaculty of Science, Radoja Domanovića 12, 34000 Kragujevac, Serbia and Montenegro |
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Abstract: | In this paper we prove that all positive eigenvalues of the Laplacian of an arbitrary simple graph have some positive lower bounds. For a fixed integer k 1 we call a graph without isolated vertices k-minimal if its kth greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every k 3 the path Pk+1 on k + 1 vertices is the unique k-minimal graph. |
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