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 with the following property: For any i and j it holds that a vertex from the i th block of the partition has exactly m i , j neighbors inside the j th block. 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 witnesses this fact. ther explore a case where the automorphism is required to exchange a given pair of edges and show some positive and negative results.