On the symmetric doubly stochastic matrices that are determined by their spectra

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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 PᵗAP 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 λ= (1,λ₂,...,λₙ) such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum λ. 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.