A046751 -id:A046751 - cp's OEIS frontend

A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 2, 5, 9, 15, 21, 24, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 2, 5, 10, 21, 41, 65, 97, 131, 148, 148, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 2, 5, 11, 24, 56, 115, 221, 402, 663, 980, 1312, 1557, 1646, 1557

Offset: 1

N. J. A. Sloane, Mar 15 1996

T(n,k)=1 for n>=2 with k=0, k=1, k=n*(n-1)/2-1 and k=n*(n-1)/2 (therefore the quadruple {1,1,1,1} marks the transition to the next sublist for a given number of vertices (n>2)). [Edited by Peter Munn, Mar 20 2021]

			Triangle begins:
1,
1,1,
1,1,1,1,
1,1,2,3,2,1,1, [graphs with 4 nodes and from 0 to 6 edges]
1,1,2,4,6,6,6,4,2,1,1,
1,1,2,5,9,15,21,24,24,21,15,9,5,2,1,1,
1,1,2,5,10,21,41,65,97,131,148,148,131,97,65,41,21,10,5,2,1,1,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 264.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 519.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 240.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, Rows 1 to 20 of triangle, flattened
Leonid Bedratyuk, A new formula for the generating function of the numbers of simple graphs, arXiv:1512.06355 [math.CO], 2015.
FindStat - Combinatorial Statistic Finder, The number of edges of a graph.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 65.
Sriram V. Pemmaraju, Combinatorica 2.0
Marko R. Riedel, Number of distinct connected digraphs
Gordon Royle, Small graphs
S. S. Skiena, Generating graphs
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Overview of the 12 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 11
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 12
James Turner and William H. Kautz, 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.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Simple Graph
A. E. Yurtsun, Principles of enumeration of the number of graphs, Ukrainian Mathematical Journal, January-February, 1967, Volume 19, Issue 1, pp 123-125, DOI 10.1007/BF01085184.

Row sums give A000088.
Cf. A046742, A046751, A000717, A001432, A001431, A001430, A001433, A001434, A048179, A048180 etc.
Cf. also A039735, A002905, A054924 (connected), A084546 (labeled graphs).
Row lengths: A000124; number of connected graphs for given number of vertices: A001349; number of graphs for given number of edges: A000664.
Cf. also A000055.

seq(seq(GraphTheory:-NonIsomorphicGraphs(v,e),e=0..v*(v-1)/2),v=1..9); # Robert Israel, Dec 22 2015

<< Combinatorica`; Table[CoefficientList[GraphPolynomial[n, x], x], {n, 8}] // Flatten (* Eric W. Weisstein, Mar 20 2013 *)
<< Combinatorica`; Table[NumberOfGraphs[v, e], {v, 8}, {e, 0, Binomial[v, 2]}] // Flatten (* Eric W. Weisstein, May 17 2017 *)
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_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/ g]^g,{j, 1, i-1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[ c-1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^#&], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x]&;
Array[row, 8] // Flatten (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

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)))}
G(n, A=0) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+A)); s/n!}
{ for(n=1, 7, print(Vecrev(G(n)))) } \\ Andrew Howroyd, Oct 22 2019, updated  Jan 09 2024

def T(n,k):
    return len(list(graphs(n, size=k)))
# Ralf Stephan, May 30 2014

O.g.f. for n-th row: 1/n! Sum_g det(1-g z^2)/det(1-g z) where g runs through the natural matrix representation of the pair group A^2_n (for A^2_n see F. Harary and E. M. Palmer, Graphical Enumeration, page 83). - Leonid Bedratyuk, Sep 23 2014

Additional comments from Arne Ring (arne.ring(AT)epost.de), Oct 03 2002

Text belonging in a different sequence deleted by Peter Munn, Mar 20 2021

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 0

N. J. A. Sloane

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 5   6;
  0, 0, 0, 1, 5, 13,  11;
  0, 0, 0, 0, 4, 19,  33,  23;
  0, 0, 0, 0, 2, 22,  67,  89,  47;
  0, 0, 0, 0, 1, 20, 107, 236, 240, 106;
  ... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.
Gordon Royle, Small graphs
Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

Main diagonal is A000055.
Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.
Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).
Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
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)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 23 2019

a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019

A054924 Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled connected graphs with n nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 4, 2, 1, 1, 0, 0, 0, 0, 0, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114

Offset: 1

N. J. A. Sloane

			Triangle begins:
1;
0,1;
0,0,1,1;
0,0,0,2,2,1,1;
0,0,0,0,3,5,5,4,2,1,1;
0,0,0,0,0,6,13,19,22,20,14,9,5,2,1,1;
the last batch giving the numbers of connected graphs with 6 nodes and from 0 to 15 edges.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, Rows 1 to 20 of triangle, flattened (corrected by Sean A. Irvine, Apr 29 2022)
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
Sean A. Irvine, Java code (github)
Gordon Royle, Small graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967

Cf. A008406, A054925.
Other versions of this triangle: A046751, A076263, A054923, A046742.
Row sums give A001349, column sums give A002905. A046751 is essentially the same triangle. A054923 and A046742 give same triangle but read by columns.
Main diagonal is A000055. Next diagonal is A001429. Largest entry in each row gives A001437.

A076263 gives a Mathematica program which produces the nonzero entries in each row.
Needs["Combinatorica`"]; Table[Print[row = Join[Array[0&, n-1], Table[ Count[ Combinatorica`ListGraphs[n, k], g_ /; Combinatorica`ConnectedQ[g]], {k, n-1, n*(n-1)/2}]]]; row, {n, 1, 8}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)

A046742 Triangle of number of connected graphs with k >= 1 edges and n nodes (2 <= n <= k+1).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 6, 0, 0, 1, 5, 13, 11, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 1

N. J. A. Sloane

			1;
0 1;
0 1 2;
0 0 2 3;
0 0 1 5 6;
0 0 1 5 13 11;
0 0 0 4 19 33 23;
0 0 0 2 22 67 89 47;
0 0 0 1 20 107 236 240 106;
0 0 0 1 14 132 486 797 657 235;
0 0 0 0 9 138 814 2075 2678 1806 551;
0 0 0 0 5 126 1169 4495 8548 8833 5026 1301;
0 0 0 0 2 95 1454 8404 22950 33851 28908 13999 3159;
0 0 0 0 1 64 1579 13855 53863 109844 130365 93569 39260 7741;
0 0 0 0 1 40 1515 20303 112618 313670 499888 489387 300748 110381 19320;
0 0 0 0 0 21 1290 26631 211866 803905 1694642 2179949 1799700 959374 311465 ...
... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 1 with 6 nodes).

Sean A. Irvine, Table of n, a(n) for n = 1..190
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
Gordon Royle, Small graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points, Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967. doi: 10.2172/4180737. Table 1 (complete up to 18 nodes)

Cf. A002905 (row sums), A008406, A046751, A054923, A054924 (transpose), A001349 (column sums).

Data corrected by Sean A. Irvine, Apr 23 2021

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A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

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A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

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A054924 Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled connected graphs with n nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

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A046742 Triangle of number of connected graphs with k >= 1 edges and n nodes (2 <= n <= k+1).

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A094602 Total number of edges in all connected unlabeled graphs on n nodes.

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