  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
$\lambda_2, \ldots, \lambda_n$ are non-positive real numbers with $1 +
\lambda_2 + \ldots + \lambda_n \geqslant 1/2$, then there exists a symmetric,
doubly stochastic matrix whose spectrum is precisely $(1, \lambda_2, \ldots,
\lambda_n)$. 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.
