A000171 Number of self-complementary graphs with n nodes.

1, 0, 0, 1, 2, 0, 0, 10, 36, 0, 0, 720, 5600, 0, 0, 703760, 11220000, 0, 0, 9168331776, 293293716992, 0, 0, 1601371799340544, 102484848265030656, 0, 0, 3837878966366932639744, 491247277315343649710080, 0, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

a(n) = A007869(n)-A054960(n), where A007869(n) is number of unlabeled graphs with n nodes and an even number of edges and A054960(n) is number of unlabeled graphs with n nodes and an odd number of edges.

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 139, Table 6.1.1.
• R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
• 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..100
• H. Fripertinger, Self-complementary graphs
• Victoria Gatt, Mikhail Klin, Josef Lauri, Valery Liskovets, From Schur Rings to Constructive and Analytical Enumeration of Circulant Graphs with Prime-Cubed Number of Vertices, in Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, (Pilsen, Czechia, WAGT 2016) Vol. 305, Springer, Cham, 37-65.
• Richard A. Gibbs, Self-complementary graphs J. Combinatorial Theory Ser. B 16 (1974), 106--123. MR0347686 (50 #188). - _N. J. A. Sloane_, Mar 27 2012
• Sebastian Jeon, Tanya Khovanova, 3-Symmetric Graphs, arXiv:2003.03870 [math.CO], 2020.
• B. D. McKay, Self-complementary graphs
• R. C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc., 38 (1963), 99-104.
• Eric Weisstein's World of Mathematics, Self-Complementary Graph
• D. Wille, Enumeration of self-complementary structures, J. Comb. Theory B 25 (1978) 143-150

Crossrefs

Cf. A047660, A051251, A047832.

Cf. A008406 (triangle of coefficients of the "graph polynomial").

Programs

Mathematica

< -1, {n, 1, 20}]  (* Geoffrey Critzer, Oct 21 2012 *)
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_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
a[n_] := Module[{s = 0}, Switch[Mod[n, 4], 2|3, 0, _, Do[s += permcount[4 p]*2^edges[p]*If[OddQ[n], n*2^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]; s/n!]];
Array[a, 40] (* Jean-François Alcover, Aug 26 2019, after Andrew Howroyd *)

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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + 2*sum(i=1, #v, v[i])}
a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p)) * 2^edges(p) * if(n%2, n*2^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 16 2018

Formula

a(4n) = A003086(2n).

a(4*n+1) = A047832(n), a(4*n+2) = a(4*n+3) = 0. - Andrew Howroyd, Sep 16 2018

Extensions

More terms from Ronald C. Read and Vladeta Jovovic