Block transitivity and degree matrices

Abstract

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 ith block of the partition has exactly mi, j neighbors inside the jth 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 witnesses this fact. We further explore a case where the automorphism is required to exchange a given pair of edges and show some positive and negative results. © 2007 Elsevier Ltd. All rights reserved.