Some results on the symmetric doubly stochastic inverse eigenvalue problem

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$‎, ‎to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$‎. ‎If there exists an $ntimes n$ symmetric doubly stochastic matrix $A$ with $sigma$ as its spectrum‎, ‎then the list $sigma$ is s.d.s‎. ‎realizable‎, ‎or such that $A$ s.d.s‎. ‎realizes $sigma$‎. ‎In this paper‎, ‎we propose a new sufficient condition for the existence of the symmetric doubly stochastic matrices with prescribed spectrum‎. ‎Finally‎, ‎some results about how to construct new s.d.s‎. ‎realizable lists from the known lists are presented.