A matrix is called "hollow" if all its diagonal entries are 0. Holllow, symmetric nonnegative matrices arise in a variety of ways, including as adjacency matrices of graphs.
The possible distribution of the eigenvalues (between positive and negative) is of interest, in particular to prove new relationships between eigenvalues and diagonal entries of
LaPlacians of graphs. In particular, we study conditions under which such matrices have few negative eigenvalues. A sample result is that the number of nonpositive eigenvalues
of adjacency matrices grows with the the number of vertices. But, there is much more to say.
