Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace.

Using this property, one can obtain a lower bound on the $k$-th largest eigenvalue of a $t \times t$ principal submatrix of $A$, using the $(k+n-t)$-th largest eigenvalue of $A$. This lower bound is best possible, for example when $A$ is diagonal. But for many interesting (fixed) matrices, such bound is usually far from being optimal. For example, let $$A=\begin{bmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 \end{bmatrix}$$

The eigenvalues of $A$ are $2, 0, 0, -2$. So if we would like to bound from below the largest eigenvalue of its $3 \times 3$ principal submatrix using Cauchy's Theorem, we only get a lower bound of $0$. However it is straightforward to check that it is always at least $\sqrt{2}$.

I am wondering if there is a more "quantitative" Interlacing Theorem, say if your matrix satisfies some additional properties (non-negative, binary, etc.), then one can obtain a better lower bound on the $k$-th largest eigenvalue of a $t \times t$ principal submatrix of $A$?