A004110 -id:A004110 - cp's OEIS frontend

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).

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.

A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

Original entry on oeis.org

0, 0, 1, 3, 28, 475, 14646, 813813, 82060392, 15251272983, 5312295240010, 3519126783483377, 4487168285715524124, 11116496280631563128723, 53887232400918561791887118, 513757147287101157620965656285, 9668878162669182924093580075565776

Offset: 0

Gus Wiseman, Sep 04 2019

nonn

A graph is covering if the vertex set is the union of the edge set, so there are no isolated vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 connected covering graphs with at least one endpoint.

The non-connected version is A327227.
The non-covering version is A327364.
Graphs with endpoints are A245797.
Connected covering graphs are A001187.
Connected graphs with bridges are A327071.
Cf. A004110, A059166, A006125, A006129, A059166, A100743, A141580, A322395, A327105, A327229, A327230, A327366, A327369, A327377.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

seq(n)={Vec(serlaplace(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

Inverse binomial transform of A327364.

a(n) = A001187(n) - A059166(n). - Andrew Howroyd, Sep 11 2019

Terms a(7) and beyond from Andrew Howroyd, Sep 11 2019

A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 10, 12, 12, 4, 3, 253, 260, 160, 60, 35, 0, 12068, 9150, 4230, 1440, 480, 66, 15, 1052793, 570906, 195048, 53200, 12600, 2310, 427, 0, 169505868, 63523656, 15600032, 3197040, 585620, 95088, 14056, 1016, 105

Offset: 0

Gus Wiseman, Sep 05 2019

A graph is covering if there are no isolated vertices.

			Triangle begins:
      1
      0     0
      0     0     1
      1     0     3     0
     10    12    12     4     3
    253   260   160    60    35     0
  12068  9150  4230  1440   480    66    15

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A006129.
Column k = 0 is A100743.
Column k = n is A123023.
Row sums without the first column are A327227.
The non-covering version is A327369.
The unlabeled version is A327372.
Cf. A004110, A006125, A055302, A055314, A059167, A245797, A327362, A327364, A327370.

Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
  my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));
  Vecrev(n!*polcoef(A, n), n + 1);
}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2019

Column-wise inverse binomial transform of A327369.

E.g.f.: exp(-x)*exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019

Terms a(28) and beyond from Andrew Howroyd, Oct 05 2019

A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 22, 48, 76, 97, 102, 84, 60, 39, 20, 10, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 25, 64, 152, 331, 617, 930, 1173, 1253, 1140

Offset: 0

N. J. A. Sloane, Nov 14 2006

			Triangle begins:
[0] 1;
[1] 1;
[2] 1, 0;
[3] 1, 0, 0, 1;
[4] 1, 0, 0, 1, 1, 1, 1;
[5] 1, 0, 0, 1, 1, 2, 4, 3,  2,  1,  1;
[6] 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1;
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

R. W. Robinson, Rows 0 through 16, flattened

Row sums are A004110.

Cf. A008406, A240168, A369928 (labeled).

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w) * y^(n-k)*polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); Vecrev(s, binomial(n,2)+1)}
{ for(n=0, 6, print(row(n))) } \\ Andrew Howroyd, Feb 07 2024

T(n,k) = A008406(n,k) - A240168(n,k). - Andrew Howroyd, Apr 16 2021

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

A327372 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 3, 1, 1, 1, 1, 11, 5, 4, 1, 2, 0, 62, 29, 18, 6, 4, 2, 1, 510, 225, 101, 32, 13, 4, 3, 0, 7459, 2674, 842, 223, 72, 19, 9, 3, 1, 197867, 50834, 10784, 2171, 504, 115, 34, 9, 4, 0, 9808968, 1653859, 228863, 32322, 5268, 944, 209, 46, 16, 4, 1

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
   1
   0  0
   0  0  1
   1  0  1  0
   3  1  1  1  1
  11  5  4  1  2  0

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Gus Wiseman, The unlabeled graphs covering 5 vertices with k endpoints.

Row sums are A002494.
Column k = 0 is A261919.
The non-covering version is A327371.
The labeled version is A327377.
Cf. A000088, A004110, A294217, A327227, A327369, A327370.

\\ Needs G(n) defined in A327371.
T(n)={my(v=Vec(G(n)*(1 - x))); vector(#v, n, Vecrev(v[n], n))}
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 11 2024

Column-wise first differences of A327371.

Terms a(21) and beyond from Andrew Howroyd, Sep 11 2019

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

cp's OEIS Frontend

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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A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

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A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

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A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).

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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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A327372 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

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