Ruling out certain 5-spectra for the symmetric nonnegative inverse eigenvalue problem

A method is developed to show that certain spectra cannot be realized for the S-NIEP. It is applied in the 5-by-5 case to rule out many spectra that were previously unresolved. These are all in the case of 3 positive and 2 negative eigenvalues as all other cases are now resolved. For spectra of the sort we discuss, a diagram is given of the spectra that are excluded here, as well as those trivially realizable, those realizable because of the trace 0 case and those that may also be excluded because of the J–L–L conditions. A small region remains unresolved; it is a very small fraction of the area of those spectra we consider.