A062734 -id:A062734 - cp's OEIS frontend

A123534 Triangular array T(n,k) giving number of 2-connected graphs with n labeled nodes and k edges (n >= 3, n <= k <= n(n-1)/2).

1, 3, 6, 1, 12, 70, 100, 45, 10, 1, 60, 720, 2445, 3535, 2697, 1335, 455, 105, 15, 1, 360, 7560, 46830, 133581, 216951, 232820, 183540, 111765, 53627, 20307, 5985, 1330, 210, 21, 1, 2520, 84000, 835800, 3940440, 10908688, 20317528

Offset: 3

N. J. A. Sloane, Nov 13 2006

			Triangle begins (n >= 3, k >= n):
  n
  3 | 1;
  4 | 3, 6, 1;
  5 | 12, 70, 100, 45, 10, 1;
  6 | 60, 720, 2445, 3535, 2697, 1335, 455, 105, 15, 1;
  ...

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

Andrew Howroyd, Rows 3 through 20, flattened (first 15 rows from R. W. Robinson)

Row sums give A013922.

Cf. A062734, A123527, A322139.

row[n_] := row[n] = Module[{s}, s = (n-1)!*Log[x/InverseSeries[#, x]& @ (x*D[#, x]& @ Log[Sum[(1+y)^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1) ])]; CoefficientList[Coefficient[s, x, n-1]/y^n, y]];
Table[row[n], {n, 3, 15}] // Flatten (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)

row(n)={Vecrev((n-1)!*polcoef(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y)^binomial(k, 2) * x^k / k!) + O(x*x^n))))), n-1)/y^n)}
{ for(n=3, 7, print(row(n))) } \\ Andrew Howroyd, Nov 30 2018

A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 68, 193, 534, 1568, 4635, 14146, 43610, 137015, 435227, 1400058, 4547768, 14917504, 49348612, 164596939, 553177992, 1872805144, 6385039022, 21917878860, 75739158828, 263438869515, 922219844982, 3249042441125, 11519128834499, 41097058489426

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

			The a(0) = 1 through a(5) = 10 simple graphs:
  {}  .  {12}  {12-13}     {12-34}        {12-13-45}
               {12-13-23}  {12-13-14}     {12-13-14-15}
                           {12-13-24}     {12-13-14-25}
                           {12-13-14-23}  {12-13-23-45}
                           {12-13-24-34}  {12-13-24-35}
                                          {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}

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

The connected case is A005703, labeled A129271.
The case of exactly n edges is A006649, covering case of A001434.
The labeled version is A369191.
Partial row sums of A370167, covering case of A008406.
The non-covering version with loops is A370168, labeled A066383.
The version with loops is A370169, labeled A369194.
The non-covering version is A370315, labeled A369192.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A054548, A062734, A116508, A367862, A367863, A368599, A369193.

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

\\ G defined in A008406.
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(8) onwards from Andrew Howroyd, Feb 19 2024

A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).

Original entry on oeis.org

1, 2, 1, 3, 6, 3, 1, 4, 18, 30, 24, 15, 6, 1, 5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1, 6, 75, 420, 1385, 3015, 4800, 6365, 7170, 6705, 5065, 3009, 1365, 455, 105, 15, 1, 7, 126, 1050, 5355, 18690, 47880, 96796, 166890, 251370, 329945, 373947, 362292, 297115

Offset: 1

Max Alekseyev, Nov 23 2006

			Triangle begins:
  1;
  2,  1;
  3,  6,   3,   1;
  4, 18,  30,  24,  15,   6,   1;
  5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1;
  ...
T(3,1) = 6 since there are three different subgraphs of K_3 with one edge and each subgraph has two connected components.

Cf. A062734.

Cf. A125206 (row-reversed version), A125207 (row sums).

{ G=sum(n=0,6,(1+y)^(n*(n-1)/2)*x^n/n!); K=G*log(G); for(n=1,6,print(Vecrev(n!*polcoeff(K,n,x)))) }

G.f.: Sum_{n,k} T(n,k)*x^n/n!*y^k=(F(x,y)-1)*exp(F(x,y)-1)=G(x,y)*log(G(x,y)) where G(x,y)=Sum_{n=0..oo} (1+y)^(n(n-1)/2)*x^n/n! and F(x,y)=1+log(G(x,y)) is g.f. of A062734.

A144228 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph has at most one cycle.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 15, 1, 10, 45, 120, 210, 222, 1, 15, 105, 455, 1365, 2913, 3670, 1, 21, 210, 1330, 5985, 20139, 49294, 68820, 1, 28, 378, 3276, 20475, 97860, 362670, 976560, 1456875, 1, 36, 630, 7140, 58905, 376236, 1914276, 7663500, 22089870, 34506640

Offset: 0

Alois P. Heinz, Sep 15 2008

			T(4,4) = 15, because there are 15 simple graphs on 4 labeled nodes with 4 edges where each maximally connected subgraph has at most one cycle:
  1-2  1-2  1-2  1-2  1-2  1-2  1 2  1 2  1-2  1 2  1 2  1-2  1-2  1-2  1 2
  |/|  |X   |/   |\|   X|   \|  |/|   X|   /|  |\|  |X   |\   | |   X   |X|
  4 3  4 3  4-3  4 3  4 3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4 3
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  3,   1;
  1,  6, 15,  20,  15;
  1, 10, 45, 120, 210, 222;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Columns k=0-3 give: A000012, A000217, A050534, A093566.
Main diagonal gives A137916.
Row sums give: A133686.
T(2n,n) gives A369828.
Cf. A000272, A057500, A007318, A000142, A129271.

cy:= proc(n) option remember; local t; 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; local j; if k=0 then 1 elif k<0 or n

t[, 0] = 1; t[n, k_] /; (k<0 || nJean-François Alcover, Jan 15 2014, after Maple *)

T(n,0) = 1, T(n,k) = 0 if k<0 or nA000272(j+1) T(n-j-1,k-j) + A057500(j+1) T(n-j-1,k-j-1)).

E.g.f.: exp(B(x,y)), where B(x,y) = Sum(Sum(A062734(n,k)*y^k*x^n/n!, k=0..n), n=1..infinity) = -1/2*log(1+LambertW(-x*y))+1/2*LambertW(-x*y) -1/4*LambertW(-x*y)^2-1/y *(LambertW(-x*y)+1/2 *LambertW(-x*y)^2). - Vladeta Jovovic, Sep 16 2008

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Original entry on oeis.org

0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

The case of an empty edge set is excluded.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500, which is also the covering case of A370317.
This is the connected case of A367862, covering A367863.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by edge count.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

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}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]

\\ Compare A370317; use A057500 for efficiency.
a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Binomial transform of A057500 (if the null graph is not connected).

a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 19 2024

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

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

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

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],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]

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

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

A094594 Total number of edges in all connected labeled graphs on n nodes.

Original entry on oeis.org

0, 1, 9, 144, 4140, 214200, 20264832, 3580049088, 1202974894656, 779257681804800, 982078160760512640, 2423077679970846226944, 11755368773275419420291072, 112487517660848696830655493120

Offset: 1

Vladeta Jovovic, Jun 06 2004

a[1]:=0: for n from 1 to 16 do a[n]:= binomial(n,2)*2^(binomial(n,2)-1)-sum(binomial(n,k)*2^binomial(n-k,2)*a[k],k=1..n-1) od: seq(a[n],n=1..16); # Emeric Deutsch, Dec 18 2004

nn=14;f[x_,y_]:=Sum[(1+y)^Binomial[n,2]x^n/n!,{n,0,nn}];Drop[Range[0,nn]!CoefficientList[Series[D[Log[f[x,y]],y]/.y->1,{x,0,nn}],x],1] (* Geoffrey Critzer, Sep 04 2013 *)

E.g.f.: A(x)/B(x), where A(x) is e.g.f. of A095351 and B(x) is e.g.f. of A006125. recurrence: a(n) = binomial(n, 2)*2^(binomial(n, 2) - 1) - Sum_{k=1..n-1} binomial(n, k)*2^binomial(n-k, 2)*a(k).

a(n) = Sum_{k=0..binomial(n,2)} A062734(n,k)*k. - Geoffrey Critzer, Sep 04 2013

More terms from Emeric Deutsch, Dec 18 2004

cp's OEIS Frontend

A123534 Triangular array T(n,k) giving number of 2-connected graphs with n labeled nodes and k edges (n >= 3, n <= k <= n(n-1)/2).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A144228 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph has at most one cycle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094594 Total number of edges in all connected labeled graphs on n nodes.

Views

Author

Keywords

Programs

Maple

Mathematica

Formula

Extensions