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A002218 Number of unlabeled nonseparable graphs (or blocks) with n nodes.
(Formerly M2873 N1155)
54
1, 1, 1, 3, 10, 56, 468, 7123, 194066, 9743542, 900969091, 153620333545, 48432939150704, 28361824488394169, 30995890806033380784, 63501635429109597504951, 244852079292073376010411280, 1783160594069429925952824734641, 24603887051350945867492816663958981 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

By definition, a(n) gives the number of graphs with zero cutpoints. - Travis Hoppe, Apr 28 2014

Isolated points (and hence the singleton graph K_1) are considered blocks; cf. West (2000, p. 155). - Eric W. Weisstein, Dec 07 2021

For n > 2, a(n) is also the number of simple biconnected graphs on n nodes. - Eric W. Weisstein, Dec 07 2021

REFERENCES

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).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..40 [terms 1..26 from R. W. Robinson; a(1) changed from 0 to 1 by Georg Fischer, Jul 11 2022]

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

PROG

(PARI) \\ 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

CROSSREFS

Column k=0 of A325111 (for n>1).

Cf. A000088, A001349, A006289, A006290, A004115, A013922, A241767.

Cf. A010355, A339070, A339071.

Sequence in context: A013009 A301920 A203416 * A107871 A111270 A307906

Adjacent sequences: A002215 A002216 A002217 * A002219 A002220 A002221

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from R. C. Read (rcread(AT)math.uwaterloo.ca). Robinson and Walsh list the first 26 terms.

a(1) changed from 0 to 1 by Eric W. Weisstein, Dec 07 2021

STATUS

approved

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Last modified December 5 19:57 EST 2022. Contains 358588 sequences. (Running on oeis4.)