|
|
REFERENCES
|
Benjamin A. Blumer, Michael S. Underwood and David L. Feder, Single-qubit unitary gates by graph scattering, arXiv:1111.5032, 2011
Gunnar Brinkmann, Kris Coolsaet, Jan Goedgebeur and Hadrien Melot, House of Graphs: a database of interesting graphs, Arxiv preprint arXiv:1204.3549, 2012. - From N. J. A. Sloane, Oct 08 2012
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron and C. R. Johnson, The number of equivalence patterns of symmetric sign patterns, Discr. Math., 306 (2006), 3074-3077.
R. L. Davis, The numbers of structures of finite relations, Proc. Amer. Math. Soc., 4 (1953), 486-494.
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.
P. Hegarty, On the notion of balance in social network analysis, arXiv preprint arXiv:1212.4303, 2012. - From N. J. A. Sloane, Feb 02 2013
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
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.
A. Milicevic and N. Trinajstic, "Combinatorial Enumeration in Chemistry", Chem. Modell., Vol. 4, (2006), pp. 405-469.
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.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
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..75 [From link below]
Natalie Arkus, Vinothan N. Manoharan, Michael P. Brenner. Deriving Finite Sphere Packings, Nov 24, 2010. (See Table 1.)
Keith M. Briggs, Combinatorial Graph Theory [Gives first 140 terms]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 105
E. Friedman, Illustration of small graphs
Harald Fripertinger, Graphs
S. Hougardy, Home Page
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
Brendan McKay, Maple program.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Marko Riedel, Nonisomorphic graphs.
S. S. Skiena, Generating graphs
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Simple Graph
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Degree Sequence
Index entries for "core" sequences
|
|
|
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 (vladeta(AT)eunet.rs) 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 (keith.briggs(AT)bt.com), Oct 24 2005
a(n) = a(n, 2) where a(n, t), the number of t-uniform hypergraphs on n unlabeled nodes (cf. A000665 for t = 3 and A051240 for t = 4), is a(n, t) = (sum on c: 1*c_1+2*c_2+...+n*c_n= n) per(c)*2^f(c), where per(c) = 1/(prod on i=1 to n) c_i!*i^c_i and f(c) = (1/ord(c)) * (sum on r=1 to ord(c)) (sum on x: 1*x_1+2*x_2...+t*x_t=t) (prod on k = 1 to t) binom(y(r, k; c), x_k), where ord(c) = lcm{i : c_i > 0} and y(r, k; c) = (sum on s|r with gcd(k, r/s) = 1) s*c_(k*s) (= the number of k-cycles of the rth power of a permutation of type c). - David Pasino (davepasino(AT)yahoo.com), Jan 31 2009
|
