Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues.
I think the multiplicity of $\mu_2$ can not be large.
For example I want to prove that:
If the multiplicity of $\mu_2$ in a tree $T$ be greater than $n/4$, then $T$ is the star $K_{1,n-1}$.
Any help can be useful for me.