how to prove multiplicity of the $\mu _2$ is one

Question

graph $G$ is tree and $\mu _{2}$ is the second small eigenvalue of laplacian matrix. if there exist eigenvector of $\mu _2$ such that all component of eigenvector is positive. how to prove multiplicity of the $\mu _2$ is one .

Answer 1

There's something that bothers me, and it is the following:

the laplacian matrix of a graph is semidefinite positive and singular. In particular $\mathbf e = (1,1,\dots,1)$ is one eigenvector for the null space.

If $\mu_2$ is not zero, then its eigenvector must be orthogonal to $\mathbf e$, and in particular, it cannot have all the components positive (or even nonnegative).

The existence of such an eigenvector means that $\mu_2=0$, but then the null space has dimension at least 2, so $\mu_2$ cannot have multiplicity 1.