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Number of labeled bi-point-determining graphs with n vertices.
5

%I #19 May 07 2021 20:28:27

%S 1,1,0,0,12,312,13824,1147488,178672128,52666091712,29715982846848,

%T 32452221242518272,69259424722321036032,291060255757818125657088,

%U 2421848956937579216663491584,40050322614433939228627991906304,1319551659023608317386779165849208832

%N Number of labeled bi-point-determining graphs with n vertices.

%C A bi-point determining graph is a graph in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

%D R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.

%H Andrew Howroyd, <a href="/A129583/b129583.txt">Table of n, a(n) for n = 0..50</a>

%H Ira M. Gessel and Ji Li, <a href="https://doi.org/10.1016/j.jcta.2010.03.009">Enumeration of point-determining Graphs</a>, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

%F E.g.f.: G(2*log(1+x)-x) where G(x) is the e.g.f. of A006125.

%o (PARI) seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(subst(g, x, 2*log(1+x+O(x*x^n))-x)))} \\ _Andrew Howroyd_, May 06 2021

%Y Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

%K nice,nonn

%O 0,5

%A Ji Li (vieplivee(AT)hotmail.com), May 07 2007

%E a(0)=1 prepended and terms a(13) and beyond from _Andrew Howroyd_, May 06 2021