A054924 -id:A054924 - cp's OEIS frontend

A049334 Triangle read by rows: T(n, k) is the number of unlabeled connected planar simple graphs with n >= 1 nodes and 0<=k<=3*n-6 edges.

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 4, 2, 1, 0, 0, 0, 0, 0, 6, 13, 19, 22, 19, 13, 5, 2, 0, 0, 0, 0, 0, 0, 11, 33, 67, 107, 130, 130, 96, 51, 16, 5, 0, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 804, 1112, 1211, 1026, 626, 275, 72, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Brendan McKay

Planar graphs with n >= 3 nodes have at most 3*n-6 edges.

			n\k 0  1  2  3  4  5  6  7  8  9 10 11 12
--:-- -- -- -- -- -- -- -- -- -- -- -- --
1:  1
2:  0  1
3:  0  0  1  1
4:  0  0  0  2  2  1  1
5:  0  0  0  0  3  5  5  4  2  1
6:  0  0  0  0  0  6 13 19 22 19 13  5  2

Georg Grasegger, Table of n, a(n) for n = 1..235 (rows 1..13) (terms n = 1..147 (rows 1..11) from Andrew Howroyd)
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463. (MR0068198) See page 457, equation (2.9).

Row sums are A003094.
Column sums are A046091.
Cf. A000055, A039735, A049336, A049337, A054924, A288265, A343873 (transpose).

geng -c $n $k:$k | planarg -q | countg -q # Georg Grasegger, Jul 11 2023

T(n, n-1) = A000055(n) and Sum_{k} T(n, k) = A003094(n) if n>=1. - Michael Somos, Aug 23 2015

log(1 + B(x, y)) = Sum{n>0} A(x^n, y^n) / n where A(x, y) = Sum_{n>0, k>=0} T(n,k) * x^n * y^k and similarly B(x, y) with A039735. - Michael Somos, Aug 23 2015

A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3n/2) <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 2, 1, 1, 3, 14, 25, 31, 28, 17, 9, 5, 2, 1, 1, 4, 24, 101, 254, 413, 475, 426, 306, 187, 103, 52, 23, 11, 5, 2, 1, 1, 19, 204, 1068, 3348, 7152, 11199, 13683, 13604, 11374, 8203, 5216, 2963, 1536, 737, 333, 144, 62, 25, 11, 5, 2, 1, 1

Offset: 4

Andrew Howroyd, Nov 24 2020

			Triangle begins:
===========================================================
n/k | 6  7  8   9  10 11  12  13  14  15  16 17 18 19 20 21
----+------------------------------------------------------
  4 | 1;
  5 |       1,  1,  1;
  6 |           2,  3, 4,  4,  2,  1,  1;
  7 |                  3, 14, 25, 31, 28, 17, 9, 5, 2, 1, 1;
  8 |                      4, 24 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 4..732 (rows n=4..18, extracted from Robinson's tables)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table VI, pages 19-27.

Row sums are A006290.
Column sums are A338511.
Cf. A054924, A123542, A339071.

A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1, 0, 1, 4, 20, 50, 82, 94, 81, 59, 38, 20, 10, 5, 2, 1, 1, 0, 1, 6, 40, 161, 429, 780, 1076, 1197, 1114, 885, 622, 386, 215, 112, 55, 24, 11, 5, 2, 1, 1, 0, 1, 7, 70, 433, 1729, 4796

Offset: 1

Andrew Howroyd, Nov 23 2020

			Triangle T(n,k) begins:
======================================================
n/k | 0  1  2  3  4  5  6  7  8   9  10 11 12 13 14 15
----+-------------------------------------------------
  1 | 0;
  2 |    1;
  3 |       0, 1;
  4 |          0, 1, 1, 1;
  5 |             0, 1, 2, 3, 2,  1,  1;
  6 |                0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..576 (rows 1..16, extracted from Robinson's tables)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]

Row sums are A002218.
Column sums are A010355.
Cf. A054923, A054924, A123534, A339070 (transpose), A339072.

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

A046751 Triangle read by rows of number of connected graphs with n nodes and k edges (n >= 2, 1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 0, 3, 5, 5, 4, 2, 1, 1, 0, 0, 0, 0, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 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, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114

Offset: 2

N. J. A. Sloane

			1;
0,1,1;
0,0,2,2,1, 1;
0,0,0,3,5, 5, 4, 2,  1,  1;
0,0,0,0,6,13,19,22, 20, 14,  9,  5, 2, 1, 1;
0,0,0,0,0,11,33,67,107,132,138,126,95,64,40,21,10,5,2,1,1;
[ the 4th row giving the numbers of connected graphs with 4 nodes and from 1 to 10 edges ].

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
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.)

See A054924, which is the main entry for this triangle.

Cf. A002905, A008406, A046742, A054923, A054924.

More terms from Vladeta Jovovic, Apr 21 2000

A001437 Number of connected graphs with n nodes and ceiling(n(n-1)/4) edges.

Original entry on oeis.org

1, 1, 2, 5, 22, 138, 1579, 33366, 1348674, 105925685, 15968704512, 4520384306832, 2402814904220039, 2425664021535713098, 4647586298937784001491, 16787189663016572148130262, 114716901953374968257425111039

Offset: 2

N. J. A. Sloane

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).

Sean A. Irvine, Table of n, a(n) for n = 2..40
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.

Largest entries in the rows of the table in A054924.

More terms from Sean A. Irvine, Jul 24 2012

A076263 Triangle read by rows: T(n,k) = number of nonisomorphic connected graphs with n vertices and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 5, 4, 2, 1, 1, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1, 47, 240, 797, 2075, 4495

Offset: 1

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

The index of the T(n,k) in the sequence is ((n-2)^3 - n + 6*k + 8)/6.

T(n,k)=1 for k = n*(n-1)/2-1 and k = n*(n-1)/2 (therefore {1,1} separates sublists for given numbers of vertices (n > 2)).

			There are 2 connected graphs with 4 vertices and 3 edges up to isomorphy (first graph: ((1,2),(2,3),(3,4)); second graph: ((1,2),(1,3),(1,4))). Index within the sequence is ((4-2)^3 - 4 + 6*3 + 8)/6 = 5.
Triangle begins:
   1;
   1;
   1,  1;
   2,  2,  1,   1;
   3,  5,  5,   4,   2,   1,   1;
   6, 13, 19,  22,  20,  14,   9,  5,  2,  1,  1;
  11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1;

T. D. Noe, Rows 1 to 16 of triangle, flattened (from Gordon Royle's website)
Keith M. Briggs, Combinatorial Graph Theory.
Sriram V. Pemmaraju, The Combinatorica Project
Marko R. Riedel, Number of distinct connected digraphs, Math StackExchange.
Marko Riedel, Maple code for sequences A008406 and A076263 including cycle indices.
Eric Weisstein's World of Mathematics, Connected Graph.

Row lengths (excluding first row): A000124. Number of connected graphs for given number of vertices: A001349. Number of connected graphs for given number of edges: A002905.
Number of entries in the n-th row is A152947. Row sums give A001349.
Starting each row from k=0 gives A054924, which is the main entry for this triangle.

NumberOfConnectedGraphs[vertices_, edges_] := Plus @@ ConnectedQ /@ ListGraphs[vertices, edges] /. {True->1, False ->0}
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[Plus @@ ConnectedQ /@ ListGraphs[Vert, i] /. {True -> 1, False -> 0}, {Vert, 8}, {i, Vert - 1, Vert*(Vert - 1)/2}]

Corrected by Keith Briggs and Robert G. Wilson v, May 01 2005
Rows 5, 6 & 7 from Robert G. Wilson v, Jun 21 2005
More terms from Keith Briggs, Jun 28 2005
Name corrected by Andrey Zabolotskiy, Nov 20 2017

A054926 Number of connected unlabeled graphs with n nodes and floor(n*(n-1)/4) edges.

Original entry on oeis.org

1, 0, 0, 2, 5, 19, 132, 1579, 33366, 1343120, 105723785, 15968704512, 4520384306832, 2402302590759788, 2425409960013204929, 4647586298937784001491, 16787189663016572148130262, 114715448859703502223876433517

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

a(n) = A054924(n, floor(n*(n-1)/4) ).

Description corrected by David Wasserman, Mar 05 2002

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

Original entry on oeis.org

0, 1, 5, 25, 130, 951, 9552, 160220, 4756703, 264964172, 27746801125, 5419696866001, 1964101824992259, 1319988856541150115, 1648566523004692022634, 3838409698542815296758222, 16719797018733808721401666187, 136732968429033400292302529059213

Offset: 1

Vladeta Jovovic, Jun 06 2004

nonn

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

Cf. A046751, A054923, A054924, A086314, A095351.

\\ See A054923 for G, InvEulerMT.
a(n)={subst(deriv(InvEulerMT(vector(n, k, G(k,y)))[n]), y, 1)} \\ Andrew Howroyd, Feb 01 2020

a(n) = Sum_{k=1..binomial(n,2)} k*A054924(n, k). - Andrew Howroyd, Feb 01 2020

Terms a(17) and beyond from Andrew Howroyd, Feb 01 2020

cp's OEIS Frontend

A049334 Triangle read by rows: T(n, k) is the number of unlabeled connected planar simple graphs with n >= 1 nodes and 0<=k<=3*n-6 edges.

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A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n*(n-1)/2).

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A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= 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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A046751 Triangle read by rows of number of connected graphs with n nodes and k edges (n >= 2, 1 <= k <= n(n-1)/2).

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A001437 Number of connected graphs with n nodes and ceiling(n(n-1)/4) edges.

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A076263 Triangle read by rows: T(n,k) = number of nonisomorphic connected graphs with n vertices and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

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A054926 Number of connected unlabeled graphs with n nodes and floor(n*(n-1)/4) edges.

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

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A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3n/2) <= k <= n(n-1)/2).