A000088 -id:A000088 - cp's OEIS frontend

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

0, 0, 0, 0, 7, 381, 21853, 1790135, 250562543, 66331467215, 34507857686001, 35645472109753873, 73356936892660012513, 301275024409580265134121, 2471655539736293803311467943, 40527712706903494712385171632959, 1328579255614092966328511889576785109

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{3,4}}
  {{1,2},{1,3},{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The connected case is A140638, unlabeled A140636.
The non-covering case is A367867.
The complement is A367869, connected A129271, non-covering A133686.
The version for set-systems is A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006129(n) - A367869(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A140637 Number of unlabeled graphs of positive excess with n nodes.

Original entry on oeis.org

0, 0, 0, 2, 15, 110, 936, 12073, 273972, 12003332, 1018992968, 165091159269, 50502031331411, 29054155657134165, 31426485969804026075, 64001015704527557101231, 245935864153532932681481794, 1787577725145611700547871854870, 24637809253125004524383007473440146

Offset: 1

Washington Bomfim, May 21 2008

nonn

We can find in "The Birth of the Giant Component" p. 53, see the link, the following: "The excess of a graph or multigraph is the number of edges plus the number of acyclic components, minus the number of vertices."
If G has just one complex component with 4 nodes, the "non-complex part" of G can be,
a) One forest of order 4. There are 6 forests, so 2*6=12 graphs.
b) One triangle and one isolated vertex, or 2*1=2 graphs.
c) One unicyclic graph of order 4, so 2*2=4 graphs.
Also the number of unchoosable unlabeled graphs with up to n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The labeled version is A367867, covering A367868, connected A140638. - Gus Wiseman, Feb 13 2024

			Below we show that a(8) = 12073. Note that A140636(n) is the number of connected graphs of positive excess with n nodes.
Let G be a disconnected graph of positive excess with 8 nodes. In this case, G has one or two complex components. We have 3 graphs of order 8 with two complex components. One of those graphs is depicted in the figure below:
  O---O...O---O
  |\..|...|\./|
  |.\.|...|.X.|
  |..\|...|/.\|
  O---O...O---O
If G has one complex component with 5 nodes, the non-complex part of G can be,
a) One forest of order 3. There are 3 forests, so A140636(5) * 3 = 39 graphs.
b) One triangle, so A140636(5) = 13 graphs.
If G has one complex component with 6 nodes, the non-complex part of G is a forest of order 2. There are 2 forests. We have A140636(6) * 2, or 186 graphs.
If G has one complex component with 7 nodes, the non-complex part of G is one isolated vertex. We have A140636(7), or 809 graphs.
Finally if G is connected, we have A140636(8), or 11005 graphs.
The total is 3 + 12 + 2 + 4 + 39 + 13 + 186 + 809 + 11005 = 12073.

Svante Janson, Donald E. Knuth, Tomasz Łuczak and Boris Pittel, The Birth of the Giant Component.

Cf. A000088, A005195, A001429.
The labeled complement is A133686, covering A367869, connected A129271.
The complement is A134964, connected A005703.
The connected covering case is A140636.
The labeled version is A367867, covering A367868, connected A140638.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
For multiset partitions we have A368097, complement A368098.
Factorizations of this type are A368413, complement A368414.
Cf. A001349, A002494, A006125, A055621, A367769, A367770, A367901.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

a(n) = A000088(n) - A134964(n).

A005470 Number of unlabeled planar simple graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 11, 33, 142, 822, 6966, 79853, 1140916, 18681008, 333312451

Offset: 0

N. J. A. Sloane

Euler transform of A003094. - Christian G. Bower

			a(2) = 2 since o o and o-o are the two planar simple graphs on two nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Trotter, ed., Planar Graphs, Vol. 9, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., 1993.
Turner, James; Kautz, William H. A survey of progress in graph theory in the Soviet Union. SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 19. - N. J. A. Sloane, Apr 08 2014
Vetukhnovskii, F. Ya. "Estimate of the Number of Planar Graphs." In Soviet Physics Doklady, vol. 7, pp. 7-9. 1962. - From N. J. A. Sloane, Apr 08 2014
R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

G. Brinkmann, and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007) 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
E. Friedman, Illustration of small graphs
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Planar Graph
Index entries for "core" sequences

Cf. A003094 (connected planar graphs), A034889, A039735 (planar graphs by nodes and edges).

Cf. A126201.

A003094 = Cases[Import["https://oeis.org/A003094/b003094.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A003094] (* Jean-François Alcover, Apr 25 2013, updated Mar 17 2020 *)

n=8 term corrected and n=9..11 terms calculated by Brendan McKay
Terms a(0) - a(10) confirmed by David Applegate and N. J. A. Sloane, Mar 09 2007
a(12) added by Vaclav Kotesovec after A003094 (computed by Brendan McKay), Dec 06 2014

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

cy:= proc(n) option remember;
       binomial(n-1, 2)*add((n-3)!/(n-2-t)!*n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember; `if`(k=0, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A005142 Number of connected bipartite graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 17, 44, 182, 730, 4032, 25598, 212780, 2241730, 31193324, 575252112, 14218209962, 472740425319, 21208887576786, 1286099113807999, 105567921675718772, 11743905783670560579, 1772771666309380358809, 363526952035325887859823, 101386021137641794979558045

Offset: 0

N. J. A. Sloane

Also, the number of unlabeled connected bicolored graphs having n nodes; the color classes may be interchanged. - Robert W. Robinson
Also, for n>1, number of connected triangle-free graphs on n nodes with chromatic number 2. - Keith Briggs, Mar 21 2006 (cf. A116079).
Also, first diagonal of triangle in A126736.
EULER transform of [1, 1, 1, 3, 5, 17, ...] is A033995 [1, 2, 3, 7, 13, ...]. - Michael Somos, May 13 2019

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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
Jean-François Alcover, Mathematica program
Keith M. Briggs, Combinatorial Graph Theory
CombOS - Combinatorial Object Server, Generate graphs
P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Bipartite Graph.
Eric Weisstein's World of Mathematics, n-Chromatic Graph
Eric Weisstein's World of Mathematics, n-Colorable Graph
Jonathan Wurtz and Danylo Lykov, The fixed angle conjecture for QAOA on regular MaxCut graphs, arXiv:2107.00677 [quant-ph], 2021.

Cf. A033995, A116079, A318869, A318870.

Mathematica
```
(* See the links section. *)
```

a(2*n+1) = A318870(2*n+1)/2, a(2*n) = (a(n) + A318869(n) + A318870(2*n) - A318870(n))/2. - Andrew Howroyd, Sep 04 2018

More terms from Ronald C. Read
a(0)=1 prepended by Max Alekseyev, Jun 24 2013
Terms a(21) and beyond from Andrew Howroyd, Sep 04 2018

A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Loic Foissy, Claudia Malvenuto, Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, personal communication.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
L. Foissy, C. Malvenuto, and F. Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Misha Gavrilovich and Misha Rabinovich, The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms, 2024. See p. 11.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From _N. J. A. Sloane_, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Wikipedia Topological space
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.

a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(16) from Brinkmann's and McKay's paper, sent by Vladeta Jovovic, Jan 04 2006

A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 78, 588, 8047, 205914, 10014882, 912908876, 154636289460, 48597794716736, 28412296651708628, 31024938435794151088, 63533059372622888758054, 244916078509480823407040988, 1783406527599529094009748567708, 24605674623474428415849066062642456

Offset: 0

N. J. A. Sloane

nonn

a(n) is also the number of unlabeled mating graphs with n nodes. A mating graph has no two vertices with identical sets of neighbors. - Tanya Khovanova, Oct 23 2008

F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.
F. Harary and E. Palmer, Graphical Enumeration, (1973), compare formula (8.7.11).
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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 0..26 from R. W. Robinson)
David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
R. J. Mathar, Illustrations for n=1..5 nodes.
Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
R. W. Robinson, Graphs without endpoints - computer printout.
N. J. A. Sloane, Illustration of a(0)-a(5).

Row sums of A123551.
Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A006024 (number of labeled mating graphs with n nodes), A129584 (bi-point-determining graphs).
If isolated nodes are forbidden, see A261919.
Cf. A000088.

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]];
a[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]}]; a[0] = 1;
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ Compare A000088.
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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s} \\ Andrew Howroyd, Sep 09 2018

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

Original entry on oeis.org

1, 1, 1, 2, 20, 308, 5338, 105298, 2366704, 60065072, 1702900574, 53400243419, 1836274300504, 68730359299960, 2782263907231153, 121137565273808792, 5645321914669112342, 280401845830658755142, 14788386825536445299398, 825378055206721558026931, 48604149005046792753887416

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

			Non-isomorphic representatives of the a(4) = 20 graphs:
  {}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A057500, unlabeled A001429.
Counting all vertices (not just covered) gives A116508.
The covering case is A367863, unlabeled A006649.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A003465 counts covering set-systems, unlabeled A055621, ranks A326754.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

\\ Here b(n) is A367863(n)
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A367863.

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

A005703 Number of n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012

Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

			From _Gus Wiseman_, Feb 20 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 8 graphs:
  {}  .  {12}  {12,13}     {12,13,14}     {12,13,14,15}
               {12,13,23}  {12,13,24}     {12,13,14,25}
                           {12,13,14,23}  {12,13,24,35}
                           {12,13,24,34}  {12,13,14,15,23}
                                          {12,13,14,23,25}
                                          {12,13,14,23,45}
                                          {12,13,14,25,35}
                                          {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Pseudotree
Wikipedia, Pseudoforest

Cf. A000055, A000081, A001429 (labeled A057500), A134964 (number of pseudoforests, labeled A133686).
The labeled version is A129271.
The connected complement is A140636, labeled A140638.
Non-connected: A368834 (labeled A367869) or A370316 (labeled A369191).
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.

Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
    sol = SolveAlways[
      0 == Series[
        t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
      x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021

a(n) = A000055(n) + A001429(n).

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000

a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

cp's OEIS Frontend

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

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A140637 Number of unlabeled graphs of positive excess with n nodes.

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A005470 Number of unlabeled planar simple graphs with n nodes.

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A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

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A005142 Number of connected bipartite graphs with n nodes.

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A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

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A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).

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