| Abstrakt: |
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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