Abstract
Let G be a connected graph of order n and U a unicyclic graph with the same order. We firstly give a sharp bound for mG(μ), the multiplicity of a Laplacian eigenvalue μ of G. As a straightforward result, mU(1) ⩽ n − 2. We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of mG(1) is nondecreasing. As applications, we get the distribution of mU (1) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of mU(1) ∈ {n − 5, n − 3}, the corresponding graphs U are completely determined.
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The authors are grateful to the referees for their valuable comments, corrections and suggestions, which have considerably improved the presentation of this paper.
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This work is supported by the National Natural Science Foundation of China (11961041, 11671344) and the Foundation of China Scholarship Council (201908620009).
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Wen, F., Huang, Q. On the multiplicity of Laplacian eigenvalues for unicyclic graphs. Czech Math J 72, 371–390 (2022). https://doi.org/10.21136/CMJ.2022.0499-20
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DOI: https://doi.org/10.21136/CMJ.2022.0499-20