Assume that a graph $G$ has the maximum signless Laplacian spectral radius among all $F_k$-free ($kgeq 2$) graphs of order $n$ ($ngeq 3k^2 -k-2$).

I want to prove that $G$ is connected.

On the contrary, assume that $G$ is not connected. Then we can add some new edges into $G$ so that the obtained graph $G’$ is connected and still $F_k$-free.

How can I prove that the Rayleigh quotient and the Perron-Frobenius theorem imply that $q_1 (G’)>q_1 (G)$? where $q_1 (G)$ is the signless Laplacian spectral radius of $G$.

Thanks in advance.