Say we have a graph $G$ on $n$ vertices, with eigenvalues $\lambda_1 \leq \dots \leq \lambda_n$ and spectral radius $\rho_G$. Let $H$ be the induced subgraph where we remove a single vertex from $G$, with eigenvalues $\mu_1 \leq \dots \leq \mu_{n-1}$ and spectral radius $\rho_H$.
Are there any known results which relate $\rho_G$ and $\rho_H$?
I know that the eigenvalues will interlace, so would assume at the very least that $\rho_H \leq \rho_G$ (though equally, I haven't convinced myself 100% of this, in the possible case where one of the largest eigenvalues may be negative, and the other is positive).
Aside from this, are they perhaps any other near results I could be pointed to?