Suppose we have a real, irreducible, primitive doubly stochastic matrix $A$. Further assume $A$ only has real eigenvalues. I am interested to know
Does $G(A)$, the directed graph associated with $A$, have identifiable special structures besides being strongly connected?
I think the doubly stochasticity might be irrelevant. Essentially, I am working on some doubly stochastic matrix constructed out of a directed graph and it turns out I need the eigenvalues to be real. So I am wondering whether I could connect this property with certain graph property. Or do we know certain types of digraphs have this propert?
It is well-known that if $A$ is an entry-wise irreducible nonnegative matrix, then there is a positive diagonal matrix $D$ such that $\rho ^{-1} D^{-1} A D$ is stochastic, in which $\rho$ denotes the spectral radius of $A$. Since diagonal-similarity preserves the zero-nonzero pattern, it follows that the digraphs are the same.
However, this result does not apply to doubly stochastic matrices. Thus, the digraph of a doubly stochastic matrix per your description above (in particular, possessing real eigenvalues) must possess special structure.
On the other hand, any nonnegative matrix with the same combinatorial structure will yield the same digraph — but there is no guarantee that the eigenvalues of such a matrix will be real, so perhaps this assumption is irrelevant.
Lastly, there are stochastic matrices that are similar to doubly stochastic matrices (see the 1981 LaMA paper by Charles R. Johnson).