A066383 -id:A066383 - cp's OEIS frontend

A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 16, 15, 6, 1, 0, 0, 0, 30, 135, 222, 205, 120, 45, 10, 1, 0, 0, 0, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175

Offset: 0

Vladeta Jovovic, Apr 09 2000

			From _Gus Wiseman_, Feb 14 2024: (Start)
Triangle begins:
   1
   0
   0   1
   0   0   3   1
   0   0   3  16  15   6   1
   0   0   0  30 135 222 205 120  45  10   1
Row n = 4 counts the following graphs:
  .  .  12-34  12-13-14  12-13-14-23  12-13-14-23-24  12-13-14-23-24-34
        13-24  12-13-24  12-13-14-24  12-13-14-23-34
        14-23  12-13-34  12-13-14-34  12-13-14-24-34
               12-14-23  12-13-23-24  12-13-23-24-34
               12-14-34  12-13-23-34  12-14-23-24-34
               12-23-24  12-13-24-34  13-14-23-24-34
               12-23-34  12-14-23-24
               12-24-34  12-14-23-34
               13-14-23  12-14-24-34
               13-14-24  12-23-24-34
               13-23-24  13-14-23-24
               13-23-34  13-14-23-34
               13-24-34  13-14-24-34
               14-23-24  13-23-24-34
               14-23-34  14-23-24-34
               14-24-34
(End)

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.4.

Michael De Vlieger, Table of n, a(n) for n = 0..10700
A. N. Bhavale and B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
Ashok Nivrutti Bhavale, Equivalence of labeled graphs and lattices, arXiv:2501.05064 [math.CO], 2025. See pp. 1-2, 13.
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Row sums give A006129. Cf. A054547.
The connected case is A062734, with loops A369195.
This is the covering case of A084546.
Column sums are A121251, with loops A173219.
The version with loops is A369199, row sums A322661.
The unlabeled version is A370167, row sums A002494.
A006125 counts simple graphs; also loop-graphs if shifted left.
Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

nn=5; s=Sum[(1+y)^Binomial[n,2]  x^n/n!, {n,0,nn}]; Range[0,nn]! CoefficientList[Series[ s Exp[-x], {x,0,nn}], {x,y}] //Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]],{n,0,5},{k,0,Binomial[n,2]}] (* Gus Wiseman, Feb 14 2024 *)

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(n, i)*C(C(i, 2), k), k=0...n*(n-1)/2.

E.g.f.: exp(-x)*Sum_{n>=0} (1 + y)^C(n,2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012

a(0) prepended by Gus Wiseman, Feb 14 2024

A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 18 2024

			Triangle begins:
   1
   0   1
   0   1   3   1
   0   0   6  17  15   6   1
   0   0   3  46 150 228 206 120  45  10   1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,3,12}    {1,2,3,12,13}   {1,2,3,12,13,23}
  {2,13}   {1,2,13}    {1,2,3,13}    {1,2,3,12,23}
  {3,12}   {1,2,23}    {1,2,3,23}    {1,2,3,13,23}
  {12,13}  {1,3,12}    {1,2,12,13}   {1,2,12,13,23}
  {12,23}  {1,3,23}    {1,2,12,23}   {1,3,12,13,23}
  {13,23}  {1,12,13}   {1,2,13,23}   {2,3,12,13,23}
           {1,12,23}   {1,3,12,13}
           {1,13,23}   {1,3,12,23}
           {2,3,12}    {1,3,13,23}
           {2,3,13}    {1,12,13,23}
           {2,12,13}   {2,3,12,13}
           {2,12,23}   {2,3,12,23}
           {2,13,23}   {2,3,13,23}
           {3,12,13}   {2,12,13,23}
           {3,12,23}   {3,12,13,23}
           {3,13,23}
           {12,13,23}

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

The version without loops is A054548.
This is the covering case of A084546.
Column sums are A173219.
Row sums are A322661, unlabeled A322700.
The connected case is A369195, without loops A062734.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000666, A006649, A062740, A066383, A133686, A368597, A369196, A369197.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}],Length[Union@@#]==n&]],{n,0,5},{k,0,Binomial[n+1,2]}]

T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

E.g.f.: exp(-x) * (Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

Original entry on oeis.org

1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			The a(0) = 1 through a(3) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,3}}
             {{2},{1,2}}  {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.

This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
Allowing any number of edges gives A322661, connected A062740.
The case of just pairs is A367863, covering case of A116508.
The unlabeled version is A368599.
The version contradicting strict AOC is A368730.
The connected case is A368951.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n]&]],{n,0,5}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ Andrew Howroyd, Jan 06 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2024

A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

Original entry on oeis.org

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
  .  {{1}}  {{1,2}}      {{1,2},{1,3}}
            {{1},{1,2}}  {{1,2},{2,3}}
            {{2},{1,2}}  {{1,3},{2,3}}
                         {{1},{1,2},{1,3}}
                         {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1,2},{1,3},{2,3}}

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

The minimal case is A000272.
Connected case of A066383 and A369196, loopless A369192 and A369193.
The loopless case is A129271, connected case of A369191.
The case of equality is A368951, connected case of A368597.
This is the connected case of A369194.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts (simple) graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Logarithmic transform of A368927.
From Andrew Howroyd, Feb 02 2024: (Start)
a(n) = A000169(n) + A129271(n).
E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

The minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

Inverse binomial transform of A369196.

A369192 Number of labeled simple graphs with n vertices and at most n edges (not necessarily covering).

Original entry on oeis.org

1, 1, 2, 8, 57, 638, 9949, 198440, 4791323, 135142796, 4346814276, 156713948672, 6251579884084, 273172369790743, 12969420360339724, 664551587744173992, 36543412829258260135, 2146170890448154922648, 134053014635659737513358, 8872652968135849629240560

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The version for loop-graphs is A066383, covering A369194.
The case of equality is A116508, covering A367863, also A367862.
The connected case is A129271, unlabeled A005703.
The covering case is A369191, minimal case A053530.
Counting only covered vertices gives A369193.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A143543.

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

from math import comb
def A369192(n): return sum(comb(comb(n,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 14 2024

a(n) = Sum_{k=0..n} binomial(binomial(n,2),k).

A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Feb 15 2024

			Triangle begins:
  1
  0
  0  1
  0  0  1  1
  0  0  1  2  2  1  1
  0  0  0  1  4  5  5  4  2  1  1
  0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Column sums are A000664.
Row sums are A002494.
This is the covering case of A008406, labeled A084546.
The labeled version is A054548, row sums A006129, column sums A121251.
The connected case is A054924, row sums A001349, column sums A002905.
The labeled connected case is A062734, with loops A369195.
The connected case with loops is A283755, row sums A054921.
The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
Cf. A000666, A006125, A006649, A054547, A066383, A322700.

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}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]

\\ G(n) defined in A008406.
row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

a(42) onwards from Andrew Howroyd, Feb 19 2024

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=Length[Union@@#]&]],{n,0,5}]

Binomial transform of A369194.

A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 3, 7, 19, 48, 135, 373, 1085, 3184, 9590, 29258, 90833, 285352, 908006, 2919953, 9487330, 31111997, 102934602, 343389708, 1154684849, 3912345408, 13353796977, 45906197103, 158915480378, 553897148543, 1943627750652, 6865605601382, 24411508473314, 87364180212671, 314682145679491

Offset: 0

Gus Wiseman, Feb 16 2024

nonn

			The a(0) = 1 through a(4) = 19 loop-graph edge sets (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}          {{1,2},{3,4}}
             {{1},{2}}    {{1,2},{1,3}}        {{1},{2},{3,4}}
             {{1},{1,2}}  {{1},{2},{3}}        {{1},{1,2},{3,4}}
                          {{1},{2},{1,3}}      {{1},{2,3},{2,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{3},{4}}
                          {{1},{1,2},{2,3}}    {{1,2},{1,3},{1,4}}
                          {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{2,4}}
                                               {{1},{2},{3},{1,4}}
                                               {{1},{2},{1,2},{3,4}}
                                               {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{2},{1,3},{3,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{1},{2,3},{2,4},{3,4}}
                                               {{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..50

The case of equality is A368599, covering case of A368598.
The labeled version is A369194, covering case of A066383.
This is the covering case of A370168.
The loopless version is the covering case of A370315, labeled A369192.
This is the loopless version is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000666, A005703, A014068, A062740, A116508, A368597, A368927, A369141, A369196, A369197, A369199.

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],{1,2}]], Union@@#==Range[n]&&Length[#]<=n&]]],{n,0,5}]

\\ G defined in A070166.
a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024

a(7) onwards from Andrew Howroyd, Feb 19 2024

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

Original entry on oeis.org

1, 1, 2, 8, 57, 608, 8614, 151365, 3162353, 76359554, 2088663444, 63760182536, 2147325661180, 79051734050283, 3157246719905273, 135938652662043977, 6275929675565965599, 309242148569525451140, 16197470691388774460758, 898619766673014862321176, 52639402023471657682257626

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The case of equality is A367862, covering case of A116508, also A367863.
The covering case is A369191, for loop-graphs A369194.
The version counting all vertices is A369192.
The version for loop-graphs is A369196, counting all vertices A066383.
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 graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A053530, A129271, A143543, A322661, A367916.

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

Binomial transform of A369191.

cp's OEIS Frontend

A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

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A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

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A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

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A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

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A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

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A369192 Number of labeled simple graphs with n vertices and at most n edges (not necessarily covering).

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A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

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A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.