On the multiplicity of Laplacian eigenvalues for unicyclic graphs

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 m G ( μ ), the multiplicity of a Laplacian eigenvalue μ of G . As a straightforward result, m U (1) ⩽ n − 2. We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of m G (1) is nondecreasing. As applications, we get the distribution of m U (1) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of m U (1) ∈ { n − 5, n − 3}, the corresponding graphs U are completely determined.