Block transitivity and degree matrices

  • Authors:

  • José Soto;Jiří Fiala

  • Affiliations:

  • Department of Mathematics, MIT, Cambridge, MA 02139, USA and Depto. Ing. Matemática, Universidad de Chile, Correo 3, Santiago 170-3, Chile;KAM, DIMATIA and Institute for Theoretical Computer Science (ITI), Charles University, Faculty of Mathematics and Physics, Malostranské nám.2/25, 11800 Prague, Czech Republic

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2008

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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 m"i","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.