A129586 -id:A129586 - cp's OEIS frontend

A006024 Number of labeled mating graphs with n nodes. Also called point-determining graphs.

1, 1, 1, 4, 32, 588, 21476, 1551368, 218608712, 60071657408, 32307552561088, 34179798520396032, 71474651351939175424, 296572048493274368856832, 2448649084251501449508762880, 40306353989748719650902623919616

Offset: 0

Simon Plouffe

nonn

A mating graph is one in which no two vertices have identical adjacencies with the other vertices. - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

Also number of (n-1)-node labeled mating graphs allowing loops and without isolated nodes. - Vladeta Jovovic, Mar 08 2008

			Consider the square (cycle of length 4) on vertices 1, 2, 3 and 4 in that order. Join a fifth vertex (5) to vertices 1, 3 and 4. The resulting graph is not a mating graph since vertices 1 and 3 both have the set {2, 4, 5} as neighbors. If we delete the edge (1,5) then the resulting graph is a mating graph: the neighborhood sets for vertices 1, 2, 3, 4 and 5 are respectively {2,4}, {1,3}, {2,4,5}, {1,3,5} and {3,4} - all different.

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50
I. M. Gessel and J. Li, Enumeration of Point-Determining Graphs, arXiv:0705.0042 [math.CO], 2007-2009.
I. M. Gessel and J. Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
R. C. Read, The Enumeration of Mating-Type Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)
D. Sumner, Point determination in graphs, Discrete Mathematics 5 (1973), 179-187.

Cf. A006025.
Cf. bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.
Cf. A007833, A079306 (connected)

a[n_] := Sum[StirlingS1[n, k] 2^Binomial[k, 2], {k, 0, n}];
Array[a, 15] (* Jean-François Alcover, Jul 25 2018 *)

a(n)=n!*polcoeff(sum(k=0,n,2^(k*(k-1)/2)*log(1+x+x*O(x^n))^k/k!),n) \\ Paul D. Hanna, May 20 2009

a(n) = Sum_{k=0..n} Stirling1(n, k)*2^binomial(k, 2). - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

E.g.f.: Sum_{n>=0} 2^(n(n-1)/2)*log(1+x)^n/n!. - Paul D. Hanna, May 20 2009

More terms from Ronald C. Read and Vladeta Jovovic, Feb 10 2003

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A129584 Number of unlabeled bi-point-determining graphs: graphs 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).

Original entry on oeis.org

1, 0, 0, 1, 6, 36, 324, 5280, 156088, 8415760

Offset: 1

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

This is the unlabeled case of bi-point-determining graphs, which are basically graphs that are both point-determining (no two vertices have the same neighborhoods) and co-point-determining (graphs whose complements are point-determining)

Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

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.

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

Original entry on oeis.org

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

Offset: 0

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

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).

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

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.

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

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

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

A129585 Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

Original entry on oeis.org

1, 1, 0, 0, 12, 252, 12312, 1061304, 170176656, 51134075424, 29204599254624, 32130964585236096, 68873851786953047040, 290164895151435531345024, 2417786648013402212500060416, 40014055814155246577685250570752, 1318911434129029730677931158374449664

Offset: 0

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

The calculation of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see reference.

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.

max = 15; f[x_] := x + Log[ Sum[ 2^Binomial[n, 2]*((2*Log[1 + x] - x)^n/n!), {n, 0, max}]/(1 + x)]; A129585 = Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 1](* Jean-François Alcover, Jan 13 2012, after e.g.f. *)

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

E.g.f.: 1 + x + log((Sum_{n>=0} 2^binomial(n,2)*(2*log(1+x)-x)^n/n!)/(1+x)). - Goran Kilibarda, Vladeta Jovovic, May 09 2007

E.g.f.: 1 + x + log(B(x)/(1+x)) where B(x) is the e.g.f. of A129583. - Andrew Howroyd, May 06 2021

More terms from Goran Kilibarda, Vladeta Jovovic, May 09 2007

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, May 06 2021

cp's OEIS Frontend

A006024 Number of labeled mating graphs with n nodes. Also called point-determining graphs.

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A129584 Number of unlabeled bi-point-determining graphs: graphs 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).

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

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A129585 Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

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Mathematica

PARI

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