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 A279045 Number of pairs of vertices that share no common neighbor summed over all simple labeled graphs on n nodes. 1
 0, 2, 18, 216, 4320, 155520, 10450944, 1337720832, 330225942528, 158508452413440, 148786600665415680, 274243462346494181376, 995653355660871966130176, 7136843253377130253221101568, 101189457574036357559516418539520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n)/2^binomial(n,2) is the expected number of pairs of vertices in a simple labeled graph on n nodes that share no common neighbor. This expectation approaches 0 as n gets big. Hence almost all graphs have diameter 2. REFERENCES D. B. West, Introduction to Graph Theory, Pearson, 2015, page 432. LINKS Indranil Ghosh, Table of n, a(n) for n = 1..45 FORMULA a(n) = 2^binomial(n, 2)*binomial(n, 2)*(1 - (1/2)^2)^(n - 2). EXAMPLE a(3)=18. There are 3 such pairs of vertices in the empty graph. There are 3 pairs in each of the 3 labelings of the graph with one edge. There are 2 pairs in each of the 3 labelings of the path of length two. 3 + 3*3 + 2*3 = 18. MATHEMATICA Table[2^Binomial[n, 2] Binomial[n, 2] (1 - (1/2)^2)^(n - 2), {n, 1, 15}] PROG (MAGMA) [2^Binomial(n, 2)*Binomial(n, 2)*(1-(1/2)^2)^(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 08 2016 (PARI) a(n) = 2^binomial(n, 2)*binomial(n, 2)*(1-(1/2)^2)^(n-2) \\ Indranil Ghosh, Feb 25 2017 CROSSREFS Sequence in context: A153647 A052726 A217239 * A155666 A227934 A349652 Adjacent sequences:  A279042 A279043 A279044 * A279046 A279047 A279048 KEYWORD nonn AUTHOR Geoffrey Critzer, Dec 04 2016 STATUS approved

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Last modified December 4 19:40 EST 2021. Contains 349526 sequences. (Running on oeis4.)