A001433 -id:A001433 - 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

A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 3, 4, 2, 1, 0, 1, 4, 8, 6, 3, 1, 0, 1, 5, 14, 14, 9, 3, 1, 0, 1, 7, 23, 32, 26, 12, 4, 1, 0, 1, 8, 36, 64, 66, 39, 16, 4, 1, 0, 1, 10, 54, 123, 158, 119, 60, 20, 5, 1, 0, 1, 12, 78, 219, 350, 325, 202, 83, 25, 5, 1, 0

Offset: 2

Christian G. Bower, May 09 2000

			Triangle begins:
  n=2:  1
  n=3:  1   0
  n=4:  1   1   0
  n=5:  1   1   1   0
  n=6:  1   2   2   1   0
  n=7:  1   3   4   2   1   0
  n=8:  1   4   8   6   3   1   0
  n=9:  1   5  14  14   9   3   1   0
  n=10: 1   7  23  32  26  12   4   1   0
  n=11: 1   8  36  64  66  39  16   4   1   0
  n=12: 1  10  54 123 158 119  60  20   5   1   0
  n=13: 1  12  78 219 350 325 202  83  25   5   1   0

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 80, Problem 3.9.

Andrew Howroyd, Table of n, a(n) for n = 2..1226 (rows 2..50)
Index entries for sequences related to trees

Row sums give A000055, row sums with weight k give A003228.
Columns 3 through 12: A001399(n-4), A055291, A055292, A055293, A055294, A055295, A055296, A055297, A055298, A055299.
Cf. A055300, A055301, A238416, A304222.
The labeled version is A055314.
Central column is A358107.
Left of central column is A359398.
Cf. A000088, A000272, A001349, A001433, A002494, A185650.

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
T(n)={my(u=[y]); for(n=2, n, u=concat([y], EulerMT(u))); my(r=x*Ser(u), v=Vec(r*(1-x+x*y) + (substvec(r,[x,y],[x^2,y^2]) - r^2)/2)); vector(n-1, k, Vecrev(v[1+k]/y^2, k))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

G.f.: A(x, y)=(1-x+x*y)*B(x, y)+(1/2)*(B(x^2, y^2)-B(x, y)^2), where B(x, y) is g.f. of A055277.

A358107 Number of unlabeled trees covering 2n nodes, n+1 of which are leaves.

Original entry on oeis.org

1, 1, 2, 6, 26, 119, 626, 3495, 20688, 127339, 810418, 5293790, 35351571, 240478715, 1662071181, 11646620758, 82601643511, 592110678762, 4284830131865, 31271691087861, 229980550743717, 1703097703162249, 12691879796699486, 95129358337729084, 716801612475691847

Offset: 1

Gus Wiseman, Dec 02 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Gus Wiseman, The a(4) = 6 trees covering 8 nodes, 5 of which are leaves.

Central column of A055290.
The labeled version is the central column of A055314.
For n leaves we have A359398.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unordered A001433.
Cf. A185650, A358732.

Terms a(11) and beyond from Andrew Howroyd, Jan 01 2023

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 1, 2, 8, 32, 158, 833, 4755, 28389, 176542, 1131055, 7432876, 49873477, 340658595, 2362652648, 16605707901, 118082160358, 848399575321, 6152038125538, 44981009272740, 331344933928536, 2457372361637286, 18337490246234464, 137612955519565773, 1038076541372187991

Offset: 1

Gus Wiseman, Jan 01 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Gus Wiseman, The a(4) = 8 trees covering 8 nodes, half of which are leaves.

Left of central column of A055290.
The labeled version is the left of central column of A055314.
The rooted version is A185650.
For n+1 leaves we have A358107.
The labeled version is A358732.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unlabeled A001433.

a(n) = A055290(2*n, n). - Andrew Howroyd, Jan 01 2023

Terms a(12) and beyond from Andrew Howroyd, Jan 01 2023

A006647 Number of graphs with n nodes, n-2 edges and no isolated vertices.

Original entry on oeis.org

1, 1, 3, 6, 15, 33, 83, 202, 527, 1377, 3744, 10335, 29297, 84396, 248034, 740289, 2245094, 6904206, 21522973, 67936799, 217026480, 701159919, 2289925258, 7556363054, 25184139149, 84743377436, 287815771822, 986345040471, 3409869008578

Offset: 4

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. L. Kocay, Some new methods in reconstruction theory, pp. 89 - 114 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.

Cf. A001430, A001433.

a(n) = A001430(n) - A001433(n - 1). - Sean A. Irvine, Jun 05 2017

More terms from Vladeta Jovovic, Mar 02 2008

More terms from Sean A. Irvine, Jun 05 2017

A006648 Number of graphs with n nodes, n-1 edges and no isolated vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 50, 124, 332, 895, 2513, 7172, 20994, 62366, 188696, 578717, 1799999, 5666257, 18047319, 58097540, 188953756, 620493315, 2056582095, 6877206111, 23195975865, 78891742748, 270505303760, 934890953041, 3256230606767

Offset: 2

N. J. A. Sloane

nonn

W. L. Kocay, Some new methods in reconstruction theory, pp. 89 - 114 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A001433, A001434.

a(n) = A001433(n) - A001434(n - 1). - Sean A. Irvine, Jun 06 2017

More terms from Vladeta Jovovic, Mar 02 2008

More terms from Sean A. Irvine, Jun 06 2017

A328057 Number of graphs with n nodes having fewer than n edges.

Original entry on oeis.org

1, 2, 3, 7, 14, 33, 81, 215, 601, 1808, 5721, 19133, 67218, 247377, 950679, 3806360, 15837196, 68336348, 305196782, 1408294018, 6703197359, 32861879994, 165699114887, 858237346563, 4560774579700, 24839216194151, 138505159164086, 789982051646096, 4604866422703625

Offset: 1

Sigurd Kittilsen and Lars Tveito, Oct 07 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A001433, A008406.

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[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c - 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
a[n_] := a[n] = Module[{s = O[x]^n}, Do[s += permcount[p]*edges[p, 1 + x^# + O[x]^n &], {p, IntegerPartitions[n]}]; SeriesCoefficient[s/(1-x), {x, 0, n - 1}]/n!];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 08 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)))}
a(n)={my(s=O(x^n)); forpart(p=n, s+=permcount(p)*edges(p, i->1 + x^i + O(x^n))); polcoef(s/(1-x), n-1)/n!} \\ Andrew Howroyd, Oct 22 2019

Terms a(17) and beyond from Andrew Howroyd, Oct 22 2019

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).

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A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n.

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A358107 Number of unlabeled trees covering 2n nodes, n+1 of which are leaves.

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A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

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A006647 Number of graphs with n nodes, n-2 edges and no isolated vertices.

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A006648 Number of graphs with n nodes, n-1 edges and no isolated vertices.

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Author

Keywords

References

Crossrefs

Formula

Extensions

A328057 Number of graphs with n nodes having fewer than n edges.

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Mathematica

PARI

Extensions

A243013 Number of graphs with n vertices and n-1 edges that can be gracefully labeled.

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