A129583 Number of labeled bi-point-determining graphs with n vertices.

1, 1, 0, 0, 12, 312, 13824, 1147488, 178672128, 52666091712, 29715982846848, 32452221242518272, 69259424722321036032, 291060255757818125657088, 2421848956937579216663491584, 40050322614433939228627991906304, 1319551659023608317386779165849208832

Offset: 0

Author

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

Keywords

• nice
• nonn

Comments

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).

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Crossrefs

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.

Programs

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

Formula

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

Extensions

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, May 06 2021