Suppose there are two connected graphs with $|V|=n$. One is a path graph $P$ and the other is an arbitrary tree graph $T\neq P$. If $L(G)$ is the Laplacian of the graph $G$, is it true that $$L(T) - L(P) \geq 0 ?$$
If the eigenvalues are ordered $\lambda_0 \leq \lambda_1 \leq ... \leq \lambda_{n-1}$, then previous literature gives $\lambda_0(L(T)) = \lambda_0(L(P))$, $\lambda_1(L(T)) \geq \lambda_1(L(P))$, and $\lambda_{n-1}(L(T)) \geq \lambda_{n-1}(L(P))$, which suggests this inequality might be true, but I've failed to find it anywhere in the literature.