Block transitivity and degree matrices

We say that a square matrix M of order r is a degree matrix of a given graph G if there is a so called equitable partition of its vertices into r blocks. This partition satisfies that for any i and j it holds that a vertex from the i-th block of the partition has exactly mi,j neighbors inside the j-th block. We ask whether for a given degree matrix M, there exists a graph G such that M is a degree matrix of G, and in addition, for any two edges e, f spanning between the same pair of blocks there exists an automorphism of G that sends e to f . In this work, we affirmatively answer the question for all degree matrices and show a way to construct a graph that witness this fact. We further explore a case where the automorphism is required to exchange given pair of edges and show some positive and negative results.