Relation of row sums to largest eigenvalue

Question

I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have at least $k$ zeros in them (more specifically, I know $k$ is the maximum size of an all zeros principal sub-matrix of the adjacency matrix), is there anything else I can tell about the largest eigenvalue (or others eigenvalues)?

Is there anything I can tell about the eigenvalues which implies I have that kind of principal sub-matrix?

Thanks!

Answer 1

I got an answer. This sub-matrix represents an independent set, and Haemers proved that the maximum size of an independent set is bounded above by $$ n\frac{\lambda_1\lambda_n}{\lambda_1\lambda_n-\delta^2},$$ where $\lambda_1,\lambda_n$ are the largest and smallest eigenvalues and $\delta$ is the minimum degree.