Degree distribution in random planar graphs

  • Authors:

  • Michael Drmota;Omer Giménez;Marc Noy

  • Affiliations:

  • Technische Universität Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria;Universitat Politècnica de Catalunya, Departament de Llenguatges i Sistemes Informítics, Jordi Girona 1-3, 08034 Barcelona, Spain;Universitat Politècnica de Catalunya, Departament de Matemítica Aplicada II, Jordi Girona 1-3, 08034 Barcelona, Spain

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

We prove that for each k=0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant d"k, so that the expected number of vertices of degree k is asymptotically d"kn, and moreover that @?"kd"k=1. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=@?"kd"kw^k. From this we can compute the d"k to any degree of accuracy, and derive the asymptotic estimate d"k~c@?k^-^1^/^2q^k for large values of k, where q~0.67 is a constant defined analytically.