A325111 -id:A325111 - cp's OEIS frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

0, 1, 1, 3, 10, 56, 468, 7123, 194066, 9743542, 900969091, 153620333545, 48432939150704, 28361824488394169, 30995890806033380784, 63501635429109597504951, 244852079292073376010411280, 1783160594069429925952824734641, 24603887051350945867492816663958981

Offset: 1

N. J. A. Sloane

By definition, a(n) gives the number of graphs with zero cutpoints. - Travis Hoppe, Apr 28 2014
For n > 2, a(n) is also the number of simple biconnected graphs on n nodes. - Eric W. Weisstein, Dec 07 2021
This sequence follows R. W. Robinson's definition of a nonseparable graph which includes K_2 but not the singleton graph K_1. This definition is most suited to graphical enumeration. Other authors sometimes include K_1 as a block or exclude K_2 as not 2-connected. - Andrew Howroyd, Feb 26 2023

P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 188.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
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).

Andrew Howroyd, Table of n, a(n) for n = 1..40 [terms 1..26 from R. W. Robinson]
P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp. [Annotated scanned copy]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
R. W. Robinson, Computer print-out of first 26 terms [Annotated scanned copy]
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
Andrés Santos, Density Expansion of the Equation of State, in A Concise Course on the Theory of Classical Liquids, Volume 923 of the series Lecture Notes in Physics, pp 33-96, 2016. DOI:10.1007/978-3-319-29668-5_3. See Reference 40.
Andrew J. Schultz and David A. Kofke, Fifth to eleventh virial coefficients of hard spheres, Phys. Rev. E 90, 023301, 4 August 2014
D. Stolee, Isomorph-free generation of 2-connected graphs with applications, arXiv preprint arXiv:1104.5261 [math.CO], 2011.
Rodrigo Stange Tessinari, Marcia Helena Moreira Paiva, Maxwell E. Monteiro, Marcelo E. V. Segatto, Anilton Garcia, George T. Kanellos, Reza Nejabati, and Dimitra Simeonidou, On the Impact of the Physical Topology on the Optical Network Performance, IEEE British and Irish Conference on Optics and Photonics (BICOP 2018), London.
Eric Weisstein's World of Mathematics, Biconnected Graph
Eric Weisstein's World of Mathematics, k-Connected Graph

Column k=0 of A325111 (for n>1).
Column sums of A339070.
Row sums of A339071.
The labeled version is A013922.
Cf. A000088 (graphs), A001349 (connected graphs), A006289, A006290, A004115 (rooted case), A010355 (by edges), A241767.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
{ my(N=12); Vec(OgfSeries(cycleIndexSeries(N)), -N) } \\ Andrew Howroyd, Dec 28 2020

More terms from Ronald C. Read. Robinson and Walsh list the first 26 terms.
a(1) changed from 0 to 1 by Eric W. Weisstein, Dec 07 2021
a(1) restored to 0 by Andrew Howroyd, Feb 26 2023

A241768 Number of simple connected graphs with n nodes and exactly 2 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 1, 3, 17, 101, 890, 11468, 239728

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=2 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

Original entry on oeis.org

0, 0, 1, 2, 7, 33, 244, 2792, 52448, 1690206, 96288815, 9873721048, 1841360945834, 629414405238720, 397024508142598996, 464923623652122023478, 1016016289424631486429082, 4162473006943138723685574978, 32096861904411547975392065322659

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

nonn

Terms may be computed from A004115. See formula. There is an obvious bijection between a connected graph with 1 articulation point and a multiset of at least two rooted nonseparable graphs joined at the root node. - Andrew Howroyd, Nov 24 2020

Andrew Howroyd, Table of n, a(n) for n = 1..26
Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=1 of A325111.
Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.
Cf. A004115 (rooted and without articulation points).

G.f.: x/(Product_{k>=1} (1 - x^k)^A004115(k+1)) - x - Sum_{k>=1} A004115(k)*x^k. - Andrew Howroyd, Nov 24 2020

Terms a(11) and beyond from Andrew Howroyd, Nov 24 2020

A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 32, 242, 2461, 35839

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=3 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241770 Number of simple connected graphs with n nodes and exactly 4 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 60, 527, 6056

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=4 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 9, 97, 1029

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=5 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 3, 0, 0, 10, 16, 12, 0, 0, 238, 250, 180, 60, 0, 0, 11368, 8496, 4560, 1920, 360, 0, 0, 1014888, 540568, 211680, 75600, 21000, 2520, 0, 0, 166537616, 61672192, 17186624, 4663680, 1226400, 241920, 20160, 0, 0, 50680432112, 12608406288, 2416430016, 469336896, 98431200, 20109600, 2963520, 181440, 0, 0

Offset: 0

Andrew Howroyd, Feb 23 2024

			Triangle begins:
        1;
        1,      0;
        1,      0,      0;
        1,      3,      0,     0;
       10,     16,     12,     0,     0;
      238,    250,    180,    60,     0,    0;
    11368,   8496,   4560,  1920,   360,    0, 0;
  1014888, 540568, 211680, 75600, 21000, 2520, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Columns k=0..3 are A013922(n>1), A013923, A013924, A013925.
Row sums are A001187.
Cf. A001710, A325111 (unlabeled version).

J(p, n)={my(u=Vecrev(p,1+n)); forstep(k=n, 1, -1, u[k] -= k*u[k+1]; u[k]/=n+1-k); u}
G(n)={log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))}
T(n)={my(v=Vec(serlaplace( 1 + ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ))); vector(#v, n, J(v[n], n-1))}
{ my(A=T(7)); for(i=1, #A, print(A[i])) }

T(n, n-2) = n!/2 = A001710(n) for n >= 2.

cp's OEIS Frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

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A241768 Number of simple connected graphs with n nodes and exactly 2 articulation points (cutpoints).

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A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

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A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

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A241770 Number of simple connected graphs with n nodes and exactly 4 articulation points (cutpoints).

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A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

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A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

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