Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Question

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$).

Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$.

Is there a non-trivial lower bound on $\lambda_{\max}(G)-\lambda_{\max}(G-C)$?