Degree distribution in random planar graphs - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2008

Degree distribution in random planar graphs


We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez 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 interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.
Fichier principal
Vignette du fichier
dmAI0109.pdf (234.91 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive

Dates and versions

hal-01194677 , version 1 (07-09-2015)



Michael Drmota, Omer Gimenez, Marc Noy. Degree distribution in random planar graphs. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. pp.163-178, ⟨10.46298/dmtcs.3562⟩. ⟨hal-01194677⟩


55 View
586 Download



Gmail Facebook X LinkedIn More