On the symmetric doubly stochastic matrices that are determined by their spectra and their connection with spectral graph theory

A symmetric doubly stochastic matrix is said to be determined by its spectrum (DS) if the only symmetric doubly stochastic matrices that are similar to are of the form for some permutation matrix The problem of characterizing such matrices is considered here. An 'almost' the same but a more difficult problem is the following: 'Characterize all the -tuples such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum ' In this note, some general results concerning our two problems are first obtained. Then, we completely solve these two problems for the case Some connections with spectral graph theory are then studied. Finally, concerning the general case, two open questions are posed and a conjecture is introduced.