OFFSET
0,3
COMMENTS
Euler transform of the sequence A001349.
Also, number of equivalence classes of sign patterns of totally nonzero symmetric n X n matrices.
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 430.
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.
Thomas Boyer-Kassem, Conor Mayo-Wilson, Scientific Collaboration and Collective Knowledge: New Essays, New York, Oxford University Press, 2018, see page 47.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 54.
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).
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22.
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, 1976.
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
Keith M. Briggs, Table of n, a(n) for n = 0..87 (From link below)
R. Absil and H. Mélot, Digenes: genetic algorithms to discover conjectures about directed and undirected graphs, arXiv preprint arXiv:1304.7993 [cs.DM], 2013.
Natalie Arkus, Vinothan N. Manoharan and Michael P. Brenner. Deriving Finite Sphere Packings, arXiv:1011.5412 [cond-mat.soft], Nov 24, 2010. (See Table 1.)
Leonid Bedratyuk and Anna Bedratyuk, A new formula for the generating function of the numbers of simple graphs, Comptes rendus de l'Académie Bulgare des Sciences, Tome 69, No 3, 2016, p.259-268.
Benjamin A. Blumer, Michael S. Underwood and David L. Feder, Single-qubit unitary gates by graph scattering, arXiv:1111.5032 [quant-ph], 2011.
Keith M. Briggs, Combinatorial Graph Theory [Gives first 140 terms]
Gunnar Brinkmann, Kris Coolsaet, Jan Goedgebeur and Hadrien Melot, House of Graphs: a database of interesting graphs, arXiv preprint 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.
P. J. Cameron and C. R. Johnson, The number of equivalence patterns of symmetric sign patterns, Discr. Math., 306 (2006), 3074-3077.
Gi-Sang Cheon, Jinha Kim, Minki Kim and Sergey Kitaev, On k-11-representable graphs, arXiv:1803.01055 [math.CO], 2018.
CombOS - Combinatorial Object Server, Generate graphs
Karen M. Daehli, Sarah A Obead, Hsuan-Yin Lin, and Eirik Rosnes, Improved Capacity Outer Bound for Private Quadratic Monomial Computation, arXiv:2401.06125 [cs.IR], 2024.
R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
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)
Peter Dukes, Notes for Math 422: Enumeration and Ramsey Theory, University of Victoria BC Canada (2019). See page 36.
D. S. Dummit, E. P. Dummit and H. Kisilevsky, Characterizations of quadratic, cubic, and quartic residue matrices, arXiv preprint arXiv:1512.06480 [math.NT], 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Mareike Fischer, Michelle Galla, Lina Herbst, Yangjing Long, and Kristina Wicke, Non-binary treebased unrooted phylogenetic networks and their relations to binary and rooted ones, arXiv:1810.06853 [q-bio.PE], 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 105.
E. Friedman, Illustration of small graphs
Harald Fripertinger, Graphs
Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, preprint, 2015.
Lee M. Gunderson and Gecia Bravo-Hermsdorff, Introducing Graph Cumulants: What is the Variance of Your Social Network?, arXiv:2002.03959 [math.ST], 2020.
P. Hegarty, On the notion of balance in social network analysis, arXiv preprint arXiv:1212.4303 [cs.SI], 2012.
S. Hougardy, Home Page
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Richard Hua and Michael J. Dinneen, Improved QUBO Formulation of the Graph Isomorphism Problem, SN Computer Science (2020) Vol. 1, No. 19.
A. Itzhakov and M. Codish, Breaking Symmetries in Graph Search with Canonizing Sets, arXiv preprint arXiv:1511.08205 [cs.AI], 2015-2016.
Dan-Marian Joiţa and Lorentz Jäntschi, Extending the Characteristic Polynomial for Characterization of C_20 Fullerene Congeners, Mathematics (2017), 5(4), 84.
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
Maksim Karev, The space of framed chord diagrams as a Hopf module, arXiv preprint arXiv:1404.0026 [math.GT], 2014.
J. M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
Steffen Lauritzen, Alessandro Rinaldo and Kayvan Sadeghi, On Exchangeability in Network Models, arXiv:1709.03885 [math.ST], 2017.
O. B. Lupanov, On asymptotic estimates of the number of graphs and networks with n edges, Problems of Cybernetics [in Russian], Moscow 4 (1960): 5-21.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
B. D. McKay, Maple program (used to redirect to here).
B. D. McKay, Maple program [Cached copy, with permission]
B. D. McKay, Simple graphs
A. Milicevic and N. Trinajstic, Combinatorial Enumeration in Chemistry, Chem. Modell., Vol. 4, (2006), pp. 405-469.
Angelo Monti and Blerina Sinaimeri, Effects of graph operations on star pairwise compatibility graphs, arXiv:2410.16023 [cs.DM], 2024. See p. 16.
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
E. M. Palmer, Letter to N. J. A. Sloane, no date
Marko Riedel, Nonisomorphic graphs.
Marko Riedel, Compact Maple code for cycle index, sequence values and ordinary generating function by the number of edges.
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
S. S. Skiena, Generating graphs
N. J. A. Sloane, Illustration of initial terms
Quico Spaen, Christopher Thraves Caro and Mark Velednitsky, The Dimension of Valid Distance Drawings of Signed Graphs, Discrete & Computational Geometry (2019), 1-11.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Overview of the 17 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 11
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 12
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 13
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 14
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 15
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 16
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
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. 18.
S. Uijlen and B. Westerbaan, A Kochen-Specker system has at least 22 vectors, arXiv preprint arXiv:1412.8544 [cs.DM], 2014.
Mark Velednitsky, New Algorithms for Three Combinatorial Optimization Problems on Graphs, Ph. D. Dissertation, University of California, Berkeley (2020).
Eric Weisstein's World of Mathematics, Simple Graph
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Degree Sequence
E. M. Wright, The number of graphs on many unlabelled nodes, Mathematische Annalen, December 1969, Volume 183, Issue 4, 250-253
E. M. Wright, The number of unlabelled graphs with many nodes and edges Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 1032-1034.
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.
FORMULA
a(n) = 2^binomial(n, 2)/n!*(1+(n^2-n)/2^(n-1)+8*n!/(n-4)!*(3*n-7)*(3*n-9)/2^(2*n)+O(n^5/2^(5*n/2))) (see Harary, Palmer reference). - Vladeta Jovovic and Benoit Cloitre, Feb 01 2003
a(n) = 2^binomial(n, 2)/n!*[1+2*n$2*2^{-n}+8/3*n$3*(3n-7)*2^{-2n}+64/3*n$4*(4n^2-34n+75)*2^{-3n}+O(n^8*2^{-4*n})] where n$k is the falling factorial: n$k = n(n-1)(n-2)...(n-k+1). - Keith Briggs, Oct 24 2005
From David Pasino (davepasino(AT)yahoo.com), Jan 31 2009: (Start)
a(n) = a(n, 2), where a(n, t) is the number of t-uniform hypergraphs on n unlabeled nodes (cf. A000665 for t = 3 and A051240 for t = 4).
a(n, t) = Sum_{c : 1*c_1+2*c_2+...+n*c_n=n} per(c)*2^f(c), where:
..per(c) = 1/(Product_{i=1..n} c_i!*i^c_i);
..f(c) = (1/ord(c)) * Sum_{r=1..ord(c)} Sum_{x : 1*x_1+2*x_2+...+t*x_t=t} Product_{k=1..t} binomial(y(r, k; c), x_k);
..ord(c) = lcm{i : c_i>0};
..y(r, k; c) = Sum_{s|r : gcd(k, r/s)=1} s*c_(k*s) is the number of k-cycles of the r-th power of a permutation of type c. (End)
a(n) ~ 2^binomial(n,2)/n! [see Flajolet and Sedgewick p. 106, Gross and Yellen, p. 519, etc.]. - N. J. A. Sloane, Nov 11 2013
For asymptotics see also Lupanov 1959, 1960, also Turner and Kautz, p. 18. - N. J. A. Sloane, Apr 08 2014
a(n) = G(1) where G(z) = (1/n!) Sum_g det(I-g z^2)/det(I-g z) and g runs through the natural matrix n X n 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, May 02 2015
From Keith Briggs, Jun 24 2016: (Start)
a(n) = 2^binomial(n,2)/n!*(
1+
2^( -n + 1)*n$2+
2^(-2*n + 3)*n$3*(n-7/3)+
2^(-3*n + 6)*n$4*(4*n^2/3 - 34*n/3 + 25) +
2^(-4*n + 10)*n$5*(8*n^3/3 - 142*n^2/3 + 2528*n/9 - 24914/45) +
2^(-5*n + 15)*n$6*(128*n^4/15 - 2296*n^3/9 + 25604*n^2/9 - 630554*n/45 + 25704) +
2^(-6*n + 21)*n$7*(2048*n^5/45 - 18416*n^4/9 + 329288*n^3/9 - 131680816*n^2/405 + 193822388*n/135 - 7143499196/2835) + ...),
where n$k is the falling factorial: n$k = n(n-1)(n-2)...(n-k+1), using the method of Wright 1969.
(End)
a(n) = 1/n*Sum_{k=1..n} a(n-k)*A003083(k). - Andrey Zabolotskiy, Aug 11 2020
MAPLE
# To produce all graphs on 4 nodes, for example:
with(GraphTheory):
L:=[NonIsomorphicGraphs](4, output=graphs, outputform=adjacency): # N. J. A. Sloane, Oct 07 2013
seq(GraphTheory[NonIsomorphicGraphs](n, output=count), n=1..10); # Juergen Will, Jan 02 2018
# alternative Maple program:
b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
+add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= n-> b(n$2, []):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 14 2019
MATHEMATICA
Needs["Combinatorica`"]
Table[NumberOfGraphs[n], {n, 0, 19}] (* Geoffrey Critzer, Mar 12 2011 *)
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]];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)
b[n_, i_, l_] := If[n==0 || i==1, 1/n!*2^(Function[p, Sum[Ceiling[(p[[j]]-1 )/2]+Sum[GCD[p[[k]], p[[j]]], {k, 1, j-1}], {j, 1, Length[p]}]][Join[l, Table[1, {n}]]]), Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]];
a[n_] := b[n, n, {}];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2019, after Alois P. Heinz *)
PROG
(Sage)
def a(n):
return len(list(graphs(n)))
# Ralf Stephan, May 30 2014
(PARI)
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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017
(Python)
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000088(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 02 2024
CROSSREFS
KEYWORD
core,nonn,nice
AUTHOR
EXTENSIONS
Harary gives an incorrect value for a(8); compare A007149
STATUS
approved