Laplacian eigenvalues of trees

Question

Suppose that $G$ is a graph and $L$ is its Laplacian matrix and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are are its Laplacian eigenvalues. We know that the multiplicity of $\mu_2$ in star $K_{1,n-1}$ is $n-2$. I wanted to know that is there any other trees that their multiplicity of $\mu_2$ be greater than $1$?

Answer 1

Yes, there are many of such types. Here is one such example:

$$L=\begin{bmatrix} 3 & -1 & 0 & -1 & -1 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 2 & 0 & 0 & -1\\ -1 & 0 & 0 & 0 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 1 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 1\end{bmatrix}$$ $$\text{eig}(L)=\begin{pmatrix} 0 & \mathbf{0.3820} & \mathbf{0.3820} & 1.5858 & 2.6180 & 2.6180 & 4.4142\end{pmatrix}.$$