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A tree is a connected graph without cycles, with a finite or infinite number of vertices. There are many variants of trees, according to further constraints or decorations.

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Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs

How accurate is the following statement: "For tree graphs, the multiplicity of the smallest non-zero eigenvalue $\lambda_2$ of the Laplacian is 1." If not valid, in which cases does it fail to hold? I …
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