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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 $\lambda_{\max}(G)-\lambda_{\max}(G-C)$?

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