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A001349 Number of connected graphs with n nodes.
(Formerly M1657 N0649)
1, 1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476, 50335907869219, 29003487462848061, 31397381142761241960, 63969560113225176176277, 245871831682084026519528568, 1787331725248899088890200576580, 24636021429399867655322650759681644 (list; graph; refs; listen; history; text; internal format)
The singleton graph K_1 is considered connected even though it is conventionally taken to have vertex connectivity 0. - Eric W. Weisstein, Jul 21 2020
Inverse Euler transform of A000088 but with a(0) omitted so that Sum_{k>=0} A000088(n) * x^n = Product_{k>0} (1 - x^k)^-a(k). It is debatable if there is a connected graph with 0 nodes and so a(0)=0 or better start from a(1)=1. - Michael Somos, Jun 01 2013. [As Harary once remarked in a famous paper ("Is the null-graph a pointless concept?"), the empty graph has every property, which is why a(0)=1. - N. J. A. Sloane, Apr 08 2014]
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 R. C. Read, Is the null-graph a pointless concept?, pp. 37-44 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 48, c(x). Also page 242.
Lupanov, O. B. Asymptotic estimates of the number of graphs with n edges. (Russian) Dokl. Akad. Nauk SSSR 126 1959 498--500. MR0109796 (22 #681).
Lupanov, O. B. "On asymptotic estimates of the number of graphs and networks with n edges." Problems of Cybernetics [in Russian], Moscow 4 (1960): 5-21.
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).
Robin J. Wilson, Introduction to Graph Theory, Academic Press, 1972. (But see A126060!)
N. J. A. Sloane, Table of n, a(n) for n = 0..75 [Computed using Keith Briggs's values for A000088]
Michal Adamaszek, Small flag complexes with torsion, arXiv:1208.3892 [math.CO], 2012.
C. O. Aguilar, B. Gharesifard, Graph Controllability Classes for the Laplacian Leader-Follower Dynamics, 2014. See Table 1.
Jonathan Baker, Kevin N. Vander Meulen, Adam Van Tuyl, Shedding vertices of vertex decomposable well-covered graphs, Discrete Mathematics (2018) Vol. 341, Issue 12, 3355-3369. Also arXiv:1606.04447 [math.NT], 2016.
Johannes Bausch, Felix Leditzky, Error Thresholds for Arbitrary Pauli Noise, arXiv:1910.00471 [quant-ph], 2019.
Gunnar Brinkmann, Kris Coolsaet, Jan Goedgebeur and Hadrien Melot, House of Graphs: a database of interesting graphs, arXiv:1204.3549 [math.CO], 2012.
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]
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS - Combinatorial Object Server, Generate graphs
Matt DeVos, Adam Dyck, Jonathan Jedwab, Samuel Simon, Which graphs occur as gamma-graphs?, arXiv:1810.01583 [math.CO], 2018.
J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271. (Annotated scanned copy of 3 pages)
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463.
Avraham Itzhakov, Michael Codish, Incremental Symmetry Breaking Constraints for Graph Search Problems, Ben-Gurion University of the Negev (Beer-Sheva, Israel, 2019).
X. Li, D. S. Stones, H. Wang, H. Deng, X. Liu and G, Wang, NetMODE: Network Motif Detection without Nauty, PLoS ONE 7(12): e50093. - From N. J. A. Sloane, Feb 02 2013
Steffen Lauritzen, Alessandro Rinaldo, Kayvan Sadeghi, On Exchangeability in Network Models, arXiv:1709.03885 [math.ST], 2017.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
B. D. McKay, Simple Graphs
A. Milicevic and N. Trinajstic, Combinatorial Enumeration in Chemistry, Chem. Modell., Vol. 4, (2006), pp. 405-469.
Marius Möller, Laura Hindersin, Arne Traulsen, Exploring and mapping the universe of evolutionary graphs, arXiv:1810.12807 [q-bio.PE], 2018.
Marius Möller, Laura Hindersin, Arne Traulsen, Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time, Communications Biology 2 (2019), Article number: 137.
L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
Gordon Royle, Small graphs
Yoav Spector, Moshe Schwartz, Study of potential Hamiltonians for quantum graphity, arXiv:1808.05632 [cond-mat.stat-mech], 2018.
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.
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.)
D. Stolee, Isomorph-free generation of 2-connected graphs with applications, 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, 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.
James Turner, 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. 18. - N. J. A. Sloane, Apr 08 2014
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, k-Connected Graph
Myung-Gon Yoon, Peter Rowlinson, Dragos Cvetkovic, Zoran Stanic, Controllability of multi-agent dynamical systems with a broadcasting control signal, Asian J. Control 16 (4) (2014) 1066-1072, Table 1.
For asymptotics see Lupanov 1959, 1960, also Turner and Kautz, p. 18. - N. J. A. Sloane, Apr 08 2014
G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 21*x^5 + 112*x^6 + 853*x^7 + ....
# To produce all connected graphs on 4 nodes, for example (from N. J. A. Sloane, Oct 07 2013):
L:=[NonIsomorphicGraphs](4, output=graphs, outputform=adjacency, restrictto=connected):
<<"Combinatorica`"; max = 19; A000088 = Table[ NumberOfGraphs[n], {n, 0, max}]; f[x_] = 1 - Product[ 1/(1 - x^k)^a[k], {k, 1, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x], {x, 0, max}], x]; sol = First[ Solve[ Thread[ Rest[ coes + A000088 ] == 0]]]; Table[ a[n], {n, 0, max}] /. sol (* Jean-François Alcover, Nov 24 2011 *)
terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
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_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Join[{1}, EULERi[Array[a88, terms]]] (* Jean-François Alcover, Jul 28 2018, after Andrew Howroyd *)
property=lambda G: G.is_connected()
def a(n):
return len([1 for G in graphs(n) if property(G)])
# Ralf Stephan, May 30 2014
Cf. A000088, A002218, A006290, A000719, A201922 (Multiset transform).
Row sums of A054924.
Sequence in context: A128527 A076328 A128528 * A126060 A266934 A182157
More terms from Ronald C. Read

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)