OFFSET
1,5
COMMENTS
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
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
MATHEMATICA
<<Combinatorica`; Table[GraphPolynomial[n, x]/.x -> -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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Ronald C. Read and Vladeta Jovovic
STATUS
approved