Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever λ₂,…,λ_n are non-positive real numbers with 1 + λ₂ +…+ λ_n ⩾ 1/2, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely (1, λ₂,…,λ_n). We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.