  A symmetric doubly stochastic matrix A is said to be determined by its
spectra if the only symmetric doubly stochastic matrices that are similar to A
are of the form $P^TAP$ for some permutation matrix P. The problem of
characterizing such matrices is considered here. An almost the same but a more
difficult problem was proposed by [ M. Fang, A note on the inverse eigenvalue
problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432 (2010)
2925-2927] as follows: Characterize all n-tuples $\lambda=
(1,\lambda_2,...,\lambda_n)$ such that up to a permutation similarity, there
exists a unique symmetric doubly stochastic matrix with spectrum $\lambda.$ In
this short note, some general results concerning our two problems are first
obtained. Then, we completely solve these two problems for the case n = 3. Some
connections with spectral graph theory are then studied. Finally, concerning
the general case, two open questions are posed and a conjecture is introduced.
