Editing Planar graph (section)

=== Enumeration of planar graphs ===
The [[Asymptotic analysis|asymptotic]] for the number of (labeled) planar graphs on <math>n</math> vertices is <math>g\cdot n^{-7/2}\cdot \gamma^n\cdot n!</math>, where <math>\gamma\approx 27.22687</math> and <math>g\approx 0.43\times 10^{-5}</math>.<ref>{{cite journal | last1 = Giménez | first1 = Omer | last2 = Noy | first2 = Marc | year = 2009 | title = Asymptotic enumeration and limit laws of planar graphs | journal = Journal of the American Mathematical Society | volume = 22 | issue = 2| pages = 309–329 | doi=10.1090/s0894-0347-08-00624-3| arxiv = math/0501269 | bibcode = 2009JAMS...22..309G | s2cid = 3353537 }}</ref>

Almost all planar graphs have an exponential number of automorphisms.<ref>{{cite journal | last1 = McDiarmid | first1 = Colin | last2 = Steger | first2 = Angelika | author2-link = Angelika Steger | last3 = Welsh | first3 = Dominic J.A. | year = 2005| title = Random planar graphs | journal = Journal of Combinatorial Theory, Series B | volume = 93 | issue = 2| pages = 187–205 | doi = 10.1016/j.jctb.2004.09.007 | citeseerx = 10.1.1.572.857 }}</ref>

The number of unlabeled (non-isomorphic) planar graphs on <math>n</math> vertices is between <math>27.2^n</math> and <math>30.06^n</math>.<ref>{{cite journal | last1 = Bonichon | first1 = N. | last2 = Gavoille | first2 = C. | last3 = Hanusse | first3 = N. | last4 = Poulalhon | first4 = D. | last5 = Schaeffer | first5 = G. | year = 2006 | title = Planar Graphs, via Well-Orderly Maps and Trees | journal = Graphs and Combinatorics | volume = 22 | issue = 2| pages = 185–202 | doi=10.1007/s00373-006-0647-2| citeseerx = 10.1.1.106.7456 | s2cid = 22639942 }}</ref>