@article {IOPORT.05918287,
    author       = {Drmota, Michael and Gim\'enez, Omer and Noy, Marc},
    title        = {Degree distribution in random planar graphs.},
    year         = {2011},
    journal      = {Journal of Combinatorial Theory. Series A},
    volume       = {118},
    number       = {7},
    issn         = {0097-3165},
    pages        = {2102-2130},
    publisher    = {Elsevier Science (Academic Press), San Diego, CA},
    doi          = {10.1016/j.jcta.2011.04.010},
    abstract     = {Summary: We prove that for each $k\ge 0$, 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 $\sum_k d_k= 1$. The proof uses the tools developed by {\it O. Gim\'enez} and {\it M. Noy} in their solution to the problem of the asymptotic enumeration of planar graphs [J. Am. Math. Soc. 22, No. 2, 309- -329 (2009; Zbl 1206.05019) and Counting planar graphs and related families of graphs. Huczynska, Sophie (ed.) et al., Surveys in combinatorics 2009. Papers from the 22nd British combinatorial conference, St. Andrews, UK, 2009. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Serie 365, 169--210 (2009; Zbl 1182.05059)], 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 $p(w)= \sum_k d_kw^k$. From this we can compute the $d_k$ to any degree of accuracy, and derive the asymptotic estimate $d_k\sim c\cdot k^{-1/2} q^k$ for large values of $k$, where $q\approx 0.67$ is a constant defined analytically.},
    identifier   = {05918287},
}