A092430 -id:A092430 - cp's OEIS frontend

A004108 Number of n-node unlabeled connected graphs without endpoints.

1, 1, 0, 1, 3, 11, 61, 507, 7442, 197772, 9808209, 902884343, 153723152913, 48443147912137, 28363697921914475, 30996525982586676021, 63502034385272108655525, 244852545421597419740767106, 1783161611489937453151313949442, 24603891216883233547700609793901996

Offset: 0

N. J. A. Sloane

nonn

Also number of n-node unlabeled connected mating graphs, cf. A006024 and A092430 (conjectured by Vladeta Jovovic, proved by G. Kilibarda). - Vladeta Jovovic, Oct 07 2004

F. Harary and E. Palmer, Graphical Enumeration, (1973), formula (8.7.11).
Goran Kilibarda, "Enumeration of unlabeled mating graphs", Belgrade, 2004, to be published.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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 (terms 1..26 from R. W. Robinson)
David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
Gabe Cunningham and Igor Minevich, Graph Products of Groups, arXiv:2502.05648 [math.GR], 2025. See p. 5.
Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
Goran Kilibarda, Enumeration of Unlabeled Mating Graphs, Graphs and Combinatorics, Volume 23, Number 2 / April, 2007, pp. 183-199.
R. W. Robinson, Connected graphs without endpoints - computer printout.
Eric Weisstein's World of Mathematics, Connected Graph.

Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004110 (Euler Transform, n-node unlabeled graphs without endpoints).
Cf. A006024, A092430 (n-node labeled connected mating graphs).
See also A261919.
Cf. A342556, A342557.
Counts include those for distance-critical graphs, A349402.

terms = 19;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
b[n_] := Sum[permcount[p]*2^edges[p]*Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}];
A004110 = Table[b[n], {n, 1, terms-1}];
Join[{1}, EULERi[A004110]] (* Jean-François Alcover, Jan 21 2019, after Andrew Howroyd *)

Inverse Euler transform of A004110. - Andrew Howroyd, Sep 09 2018

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.

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.

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.

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.

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

A129586 Number of unlabeled connected bi-point-determining graphs (see A129583).

Original entry on oeis.org

1, 0, 0, 1, 5, 31, 293, 4986, 151096, 8264613, 812528493, 144251345591, 46649058611515, 27744159658789435, 30603223477819571330, 63039669933956074333128, 243839768084859914114367906, 1779006737976575676931317142360, 24571827603944282248499044846893618

Offset: 1

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

The calculation of the number of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see linked paper "Enumeration of point-determining Graphs".

Martin Rubey, Table of n, a(n) for n = 1..29
Florian Fürnsinn, Moritz Gangl, and Martin Rubey, Unexpectedly, a symmetry on unlabeled graphs, arXiv:2505.03665 [math.CO], 2025. See p. 18.
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.

151096 and 8264613 from Vladeta Jovovic, May 10 2007

a(n) for n >= 11 from Martin Rubey, May 08 2025

A007833 Number of point-labeled reduced two-graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 28, 448, 18788, 1419852, 207249896, 58206408344, 31725488477648, 33830818147141904, 71068681534173472576, 295648155633330113713344, 2444510010072634827916776064, 40269686339597630128483872278656, 1323732128140903183968664175047409152

Offset: 1

Peter J. Cameron

nonn

Also number of (n-1)-node labeled mating graphs without isolated nodes, cf. A006024. - Vladeta Jovovic, Mar 23 2004

Michael De Vlieger, Table of n, a(n) for n = 1..83
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Cf. A092430 (connected).

Array[Sum[StirlingS1[#, k] 2^((k - 1) (k - 2)/2), {k, #}] &, 15] (* Michael De Vlieger, Feb 03 2018 *)

a(n) = Sum_{k=1..n} s(n, k) * 2^((k-1) * (k-2) / 2) where s(n, k) are the Stirling numbers of the first kind. - Sean A. Irvine, Feb 03 2018

A232699 Number of labeled point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 15, 135, 1875, 38745, 1168545, 50017905, 3029330745, 257116925835, 30546104308335, 5065906139629335, 1172940061645387035, 379092680506164049425, 171204492289446788997825, 108139946568584292606269025, 95671942593719946611454522225

Offset: 0

Justin M. Troyka, Nov 27 2013

nonn

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.
a(n) is always odd.
For every prime p > 2, a(n) is divisible by p for all n >= p.  It follows that, if m is odd and squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q.  A similar property appears to hold for odd prime powers, in which case it would hold for all odd numbers.

			Consider n = 3. The triangle graph is point-determining, but it is not bipartite, so it is not counted in a(3). The graph 1--2--3 is bipartite, but it is not point-determining (the vertices on the two ends have the same neighborhood), so it is also not counted in a(3). The only graph counted in a(3) is the graph *--*  * (with three possible labelings). - _Justin M. Troyka_, Nov 27 2013

Andrew Howroyd, Table of n, a(n) for n = 0..100 (terms 0..20 from Justin M. Troyka)
Ira Gessel and Ji Li, Enumeration of point-determining graphs, arXiv:0705.0042 [math.CO], 2007-2009.
Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, Combinatorial species and graph enumeration, arXiv:1312.0542 [math.CO], 2013.

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A218090 (unlabeled point-determining bipartite graphs).
Cf. A232700, A088974 (labeled and unlabeled connected point-determining bipartite graphs).

terms = 20;
CoefficientList[Sqrt[Sum[((1+x)^2^k Log[1+x]^k)/k!, {k, 0, terms}]] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)

seq(n)={my(A=log(1+x+O(x*x^n))); Vec(serlaplace(sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))))} \\ Andrew Howroyd, Sep 09 2018

From Andrew Howroyd, Sep 09 2018: (Start)
a(n) = Sum_{k=0..n} Stirling1(n,k)*A047864(k).
E.g.f: sqrt(Sum_{k=0..n} exp(2^k*log(1+x))*log(1+x)^k/k!). (End)

A232700 Number of labeled connected point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 0, 12, 60, 1320, 26880, 898800, 40446000, 2568736800, 225962684640, 27627178692960, 4686229692144000, 1104514965434200320, 361988888631722352000, 165271302775469812521600, 105278651889065640047462400, 93750696652129931568573619200

Offset: 1

Justin M. Troyka, Nov 27 2013

nonn

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.

For all n >= 3, a(n) is even. For every prime p > 2, a(n) is divisible by p for all n >= p. It follows that, if m is squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q. A similar property appears to hold for prime powers, in which case it would hold for all numbers.

			Consider n = 4.  There are 12 connected point-determining bipartite graphs on 4 vertices: the graph *--*--*--*, with 12 possible labelings. - _Justin M. Troyka_, Nov 27 2013

Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..20 from Justin M. Troyka)
Ira Gessel and Ji Li, Enumeration of point-determining graphs, arXiv:0705.0042 [math.CO], 2007-2009.
Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, Combinatorial species and graph enumeration, arXiv:1312.0542 [math.CO], 2013.

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A232699, A218090 (labeled and unlabeled point-determining bipartite graphs).
Cf. A088974 (unlabeled connected point-determining bipartite graphs).

terms = 18;
G[x_] = Sqrt[Sum[((1 + x)^2^k*Log[1 + x]^k)/k!, {k, 0, terms+1}]] + O[x]^(terms+1);
A[x_] = x + Log[1 + (G[x] - x - 1)/(1 + x)];
(CoefficientList[A[x], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)

seq(n)={my(A=log(1+x+O(x*x^n))); my(p=sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))); Vec(serlaplace(x + log(1+(p-x-1)/(1+x))))} \\ Andrew Howroyd, Sep 09 2018

E.g.f.: x + log(1 + (G(x)-x-1)/(1+x)) where G(x) is the e.g.f. of A232699. - Andrew Howroyd, Sep 09 2018

A088974 Number of (nonisomorphic) connected bipartite graphs with minimum degree at least 2 and with n vertices.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 9, 45, 160, 1018, 6956, 67704, 830392, 13539344, 288643968, 8112651795, 300974046019, 14796399706863, 967194378235406, 84374194347669628, 9856131011755992817, 1546820212559671605395

Offset: 1

Felix Goldberg (felixg(AT)tx.technion.ac.il), Oct 30 2003

nonn

The terms were computed using the program Nauty.

As shown in the Hardt et al. reference, this sequence (for n >= 3) also enumerates the connected point-determining bipartite graphs. - Justin M. Troyka, Nov 27 2013

			Consider n = 4.  There is one connected bipartite graph with minimum degree at least 2: the square graph.  Also there is one connected point-determining bipartite graph: the graph *--*--*--*. - _Justin M. Troyka_, Nov 27 2013

Brendan McKay, Nauty
Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, Combinatorial species and graph enumeration, arXiv:1312.0542 [math.CO].

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs [the latter is also unlabeled graphs w/ min. degree >= 2]).
Cf. A059167 (labeled graphs w/ min. degree >= 2).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs [the latter is also unlabeled connected graphs w/ min. degree >= 2]).
Cf. A059166 (labeled connected graphs w/ min. degree >= 2).
Cf. A232699, A218090 (labeled and unlabeled point-determining bipartite graphs).
Cf. A232700 (labeled connected point-determining bipartite graphs).

More terms from Andy Hardt, Oct 31 2012

A218090 Number of unlabeled point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 8, 17, 63, 224, 1248, 8218, 75992, 906635, 14447433, 303100595, 8415834690, 309390830222, 15105805368214, 982300491033887

Offset: 0

Andy Hardt, Oct 20 2012

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.

			Consider n = 3. The triangle graph is point-determining, but it is not bipartite, so it is not counted in a(3). The graph *--*--* is bipartite, but it is not point-determining (the vertices on the two ends have the same neighborhood), so it is also not counted in a(3). The only graph counted in a(3) is the graph *--*  *. - _Justin M. Troyka_, Nov 27 2013

Ira Gessel and Ji Li, Enumeration of point-determining graphs, arXiv:0705.0042 [math.CO]
Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, Combinatorial species and graph enumeration, arXiv:1312.0542 [math.CO].

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A232699 (labeled point-determining bipartite graphs).
Cf. A232700, A088974 (labeled and unlabeled connected point-determining bipartite graphs).

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A004108 Number of n-node unlabeled connected graphs without endpoints.

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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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A129586 Number of unlabeled connected bi-point-determining graphs (see A129583).

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A007833 Number of point-labeled reduced two-graphs with n nodes.

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A232699 Number of labeled point-determining bipartite graphs on n vertices.

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A232700 Number of labeled connected point-determining bipartite graphs on n vertices.

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