Mediated digraphs and quantum nonlocality

  • Authors:

  • G. Gutin;N. Jones;A. Rafiey;S. Severini;A. Yeo

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK;Department of Mathematics, Bristol University, University Walk, Bristol, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK;Department of Mathematics and Department of Computer Science, University of York, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK

  • Venue:
  • Discrete Applied Mathematics - Special issue: Max-algebra
  • Year:
  • 2005

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Abstract

A digraph D = (V, A) is mediated if for each pair x, y of distinct vertices of D, either xy ∈ A or yx ∈ A or there is a vertex z such that both xz, yz ∈ A. For a digraph D, Δ-(D) is the maximum in-degree of a vertex in D. The nth mediation number µ(n) is the minimum of Δ-(D) over all mediated digraphs on n vertices. Mediated digraphs and µ(n) are of interest in the study of quantum nonlocality.We obtain a lower bound f(n) for µ(n) and determine infinite sequences of values of n for which µ(n) = f(n) and µ(n) f(n), respectively. We derive upper bounds for µ(n) and prove that µ(n) = f(n)(1 + o(1)). We conjecture that there is a constant c such that µ(n) ≤ f(n) + c. Methods and results of design theory and number theory are used.